Modular Representation Theory of Finite Groups Jun.-Prof. Dr

Home , Indecomposable module, Simple module

Modular Representation Theory of Finite Groups

Jun.-Prof. Dr. Caroline Lassueur TU Kaiserslautern

Skript zur Vorlesung, WS 2020/21

(Vorlesung: 4SWS // Übungen: 2SWS)

Version: 23. August 2021 Contents

Foreword iii

Conventions iv

Chapter 1. Foundations of Representation Theory6

1 (Ir)Reducibility and (in)decomposability...... 6 2 Schur’s Lemma...... 7 3 Composition series and the Jordan-Hölder Theorem...... 8 4 The Jacobson radical and Nakayama’s Lemma...... 10 Chapter5 2.Indecomposability The Structure of and Semisimple the Krull-Schmidt Algebras Theorem ...... 1115

6 Semisimplicity of rings and modules...... 15 7 The Artin-Wedderburn structure theorem...... 18 Chapter8 3.Semisimple Representation algebras Theory and their of Finite simple Groups modules ...... 2226

9 Linear representations of ﬁnite groups...... 26 10 The group algebra and its modules...... 29 11 Semisimplicity and Maschke’s Theorem...... 33 Chapter12 4.Simple Operations modules on over Groups splitting and ﬁelds Modules...... 3436

13 Tensors, Hom’s and duality...... 36 14 Fixed and cofixed points...... 39 Chapter15 5.Inflation, The Mackey restriction Formula and induction and Clifford...... Theory 3945

16 Double cosets...... 45 17 The Mackey formula...... 47 18 Cliﬀord theory...... 48

i Skript zur Vorlesung: Modular Representation Theory WS 2020/21

ii Chapter 6. Projective Modules over the Group Algebra 51

19 Radical, socle, head...... 51 20 Projective modules...... 54 21 Projective modules for the group algebra...... 54 22 The Cartan matrix...... 58 23 Symmetry of the group algebra...... 59 Chapter24 7.Representations Indecomposable of cyclic Modules groups in positive characteristic...... 6264

25 Relative projectivity...... 64 26 Verticesp and sources...... 71 27 The Green correspondence...... 74 28 -permutation modules...... 77 Chapter29 8.Green’sp-Modular indecomposability Systems theorem...... 8082 p 30 Complete discrete valuation rings...... 82 31 Splitting -modular systems...... 85 32 Lifting idempotents...... 88 Chapter33 9.Brauer Brauer Reciprocity Characters...... 9193

34 Brauer characters...... 94 Chapter35 10.Back Blocks to decomposition matrices of ﬁnite groups...... 10498 p 36 The blocks of a ring...... 104 37 -Blocks of ﬁnite groups...... 106 38 Defect Groups...... 108 Appendix39 Brauer’s 1: Background 1st and 2ndMaterial Main Module Theorems Theory...... 1000110

A Modules, submodules, morphisms...... 1000 B Free modules and projective modules...... 1003 C Direct products and direct sums...... 1005 D Exact sequences...... 1006 E Tensor products...... 1008 AppendixF Algebras 2: The...... Language of Category Theory 10131011

G Categories...... 1013 IndexH of NotationFunctors...... 10181016 Foreword

Modular Representation Theory

This text constitutes a faithful transcript of the lecture held at the TU Kaiserslautern during the Winter Semester 2020/21 (14 Weeks, 4SWS Lecture + 2SWS Exercises). ‚ finite group theory Together with the necessary theoretical foundations the main aims of this lecture are to: ‚ representation theory of finite-dimensional algebras the provide students with a modern approach to ; group algebra of a finite group learn about the and in particular of ‚ positive ; zero establish connections between the representation theory of a finite group over a field of ‚ universal properties language of category characteristic and that over a field of characteristic ; theory consistently work with and get acquainted with the linear algebra elementary . group theory Grundlagen der Mathematik Algebraische Strukturen WeEinführung assume asin diepre-requisites Algebra bachelor-level algebra courses dealing with andCommutative Algebra ,Character such as the Theory standard of Finite lectures Groups , , and . It is also strongly recommended to have attended the lectures and prior to this lecture. Therefore, in order to complement these pre-requisites, but avoid repetitions,Character Theory the Appendix of Finite deals Groups formally with some background material on module theory, but proofs are omitted. The main results of the lecture will be recovered through a different and more general approach, thus it is formally not necessary to have attended this lecture already, but Acknowledgementit definitely brings you some intuition.

: I am grateful to Gunter Malle who provided me with the Skript of his lecture "Darstellungstheorie" hold at the TU Kaiserslautern in the WS 12/13, 13/14, 15/16 and 16/17, which I used as a basis for the development of this lecture. I am also grateful to Niamh Farrell, who shared with me her text from 2019/20 to the second part of the lecture, which I had never taken myself prior to the WS20/21. I am also grateful to Kathrin Kaiser, Helena Petri and Bernhard Böhmler who mentioned typos to me in the preliminary version of these notes. Further comments, corrections and suggestions are of course more than welcome. Conventions

Unless otherwise stated, throughout these notes we make the following general assumptions: ¨ ﬁnite

¨ associative unital all groups considered are ;

all rings considered are and (i.e. possess a neutral element for the ¨ left multiplication, denoted 1); ¨ K G KG all modules considered are modules; K if is a commutative ring and a ﬁnite group, then all -modules considered are assumed to be free of ﬁnite rank when regarded as -modules. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

v Chapter 1. Foundations of Representation Theory

representation theory of ﬁnite groups In thisSchur’s chapter Lemma we review four important module-theoretic theorems, which lie at the foundations of : The Jordan-Hölder Theorem 1. : about homomorphisms between simple modules. Nakayama’s Lemma 2. : about "uniqueness" properties of composition series. The Krull-Schmidt Theorem 3. : about an essential property of the Jacobson radical. Notation4. : about direct sum decompositions intoR indecomposable submodules.

: throughout this chapter, unless otherwise speciﬁed, we let denote an arbitrary unital and associative ring. References:

Representations and cohomology. I

[Ben98] D. J. Benson. Methods of representation. Vol. theory. 30. Cambridge Vol. I Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1998. [CR90] C. W. Curtis andGroup I. Reiner. representation theory. Part B: Modular representation. John Wiley theory & Sons, Inc., New York, 1990. [Dor72] L. Dornhoff. Representations of finite groups . Marcel Dekker, Inc., New York, 1972. [NT89] H. Nagao andAdvanced Y. Tsushima. modern algebra. 2nd ed. . Academic Press, Inc., Boston, MA, 1989. [Rot10] J. J. Rotman.A course in finite group representationProvidence, theory RI: American Mathematical Society (AMS), 2010. [Web16] P. Webb. . Vol. 161. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016. 1 (Ir)Reducibility and (in)decomposability

elementary simplicity indecomposability Submodules and direct sums of modules allow us to introduce the two main notions that will enable us to break modules in pieces in order to simplify their study: and .

6 Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Deﬁnition 1.1 (simple/irreducible module / indecomposable module / semisimple module) 7

R M reducible R U Ĺ U Ĺ M R M simple irreducible (a) An -module is called if it admits an -submodule such that 0 . R M decomposable M An -module is called , or , if it is non-zero and not reducible. M ;M M “ M ‘ M R M indecomposable (b) An -module is called if possesses two non-zero proper submodules 1 2 1 2 such that . An -module is called if it is non-zero R M completely reducible semisimple and not decomposable. R (c) An -module is called or if it admits a direct sum decomposition into simple -submodules.

OurRemark primary 1.2 goal in Chapter 1 and Chapter 2 is to investigate each of these three concepts in details.

Clearly any simple module is also indecomposable, resp. semisimple. However, the converse does Exercisenot hold 1.3 in general. pR; `; ¨q R˝ “ R R R

Prove that if is a ring, then : ˝ itself˝ maybe seen as an -module via left multiplication in , i.e. where the external compositionR ˆ R lawÝÑ isR given; pr; m byq ÞÑ r ¨ m : R˝ regular R

We call the -module. R R˝ R Prove that: I R R ô R˝{I R I R (a) theC -submodules of are precisely the left ideals of ; C R ô I R R˝ (b) is a maximal left ideal of is a simple -module, and is a minimal left ideal of is simple when regarded as an -submodule of .

2 Schur’s Lemma

simple

Schur’s Lemma is a basic result, which lets us understand homomorphisms between modules, and,Theorem more 2.1 importantly, (Schur’s Lemma endomorphisms) of such modules. V;W R

R pV q (a) Let be simple -modules. Then: V fl W R pV;W q “ (i) End is a skew-field, and K A K V A (ii) if , then Hom 0. K V ă 8 (b) If is an algebraically closed field,ApV qis “ a tλ -algebra,V | λ P K andu – K:is a simple -module such that dim , then End Id Skript zur Vorlesung: Modular Representation Theory WS 2020/21

8 Proof :

f P R pV;W qzt u R pW;V q f ‰ ùñ f Ĺ V R V t u ‰ f R (a) First, we claimW that every Hom 0 admitsf “ t u an inverseV in Hom . f Indeed, 0 ker f “ Wis a properW -submodule of and 0 f Im is a non-zero - ´ submodule of . But then, on the one hand,f ker 0 , because f is simple,P R hencepW;V q is injective, and on the other hand, Im because is simple. It follows that1 RispV alsoq surjective, hence bijective. Therefore, by ExampleR pV q QA.4V(d), is invertibleV ‰ with inverseV ‰ Hom . Now,f P (ii)R p isV qzt straightforwardu from theR pV above.q For (i), ﬁrst recall that End is a ring, which is obviously non-zero as End Id and Id 0 because 0 since it is simple. Thus, as any f P ApV q K f λ P K v P V zt u End 0 is invertible, End is a skew-ﬁeld. f λ pf ´ λ V qpvq “ f ´ λ V (b) Let End . By the assumptions on , has an eigenvalue . Let 0 be an paq eigenvector of forf ´ .λ ThenV P ApVIdq ùñ 0.f Therefore,´ λ V “ ùñId isf not“ λ invertibleV : and

pV q Ď tλ | λ P K u A IdV End Id 0 Id pV q “ tλ u – K: Hence End Id , but theA reverse inclusionV also obviously holds, so that

End Id

3 Composition series and the Jordan-Hölder Theorem

simple

From Chapter 2 on, we will assume that all modules we work with can be broken into modules inDeﬁnition the sense 3.1 of ( theComposition followingseries deﬁnition. / composition factors / composition length) M R

series ﬁltration M Let be an -module. Z (a)A (or ) of“ Mis aĎ ﬁniteM Ďchain::: Ď ofM submodulesn “ M pn P ě q :

0 1 0 composition series 0 M

Z (b)A of “ isM aĎ seriesM Ď ::: Ď Mn “ M pn P ě q

0 1 0 Mi{Mi´ 0 ď i ď n Mi{Mi´ composition factors constituents M n composition 1 1 wherelength M is simple for each 1 . The quotient modules are called the (or the ) of and the integer is called the of .

Notice that, clearly, in a composition series all inclusions are in fact strict because the quotient modules are required to be simple, hence non-zero.composition series ﬁnitely generated

Next we see that the existence of a implies that theNoetherian module is Artinian . However, the converse does not hold in general. This is explained through the fact that the existence of a composition series is equivalent to the fact that the module is both and . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Deﬁnition 3.2 (Chain conditions / Artinian and Noetherian rings and modules) 9

R M descending chain condition Artinian M “ M Ě M Ě ::: Ě Mr Ě ::: Ě t u (a) An -module is said to satisfy the D m Mm “(D.C.C.)Mm on submodulesm ě m 0 1 (or to be ) if every descending chain 0 of R M ascending chain0 condition 0 0 submodules eventually becomes stationary, i.e. such that for every . Noetherian “ M Ď M Ď ::: Ď Mr Ď ::: Ď M (b) An -module is said to satisfy theD m Mm “ Mm (A.C.C.) onm ě submodulesm (or to 0 1 be ) if every ascending chain 0 of˝ submodules R left Artinian 0left Noetherian 0 R0 eventually becomes stationary, i.e. such that for every .

R “ M Ď M Ď ::: Ď Mr “ M r P Zě R M M 0 1 0 Any series of -submodules 0 ( ) of an -module may be Z reﬁned to a composition series“ ofM Ĺ. InM addition,Ĺ ::: Ĺ ifMn “ M pn P ě q

0 1 0 0 1 1 1 “ M Ĺ M Ĺ ::: Ĺ Mm “ M pm P Zě q and S M 0 1m “ n 0 π P n 1 1 0 Mi{Mi´ – Mπpiq{Mπpiq´ ď i ď n are two composition series of , then and there exists a permutation such that 1 1 Proof : Commutative Algebrafor every 1 . In particular, the composition length is well-deﬁned.

CorollarySee 3.4 . M R

M If is an -module, then TFAE: M (a) has a composition series; M M (b) satisﬁes D.C.C. and A.C.C. on submodules;

Proof :(c) Commutativesatisﬁes D.C.C. Algebra on submodules and every submodule of is ﬁnitely generated.

TheoremSee 3.5 (Hopkins’ Theorem. ) M M If is a module over a left Artinian ring, then TFAE: M (a) has a composition series; M (b) satisﬁes D.C.C. on submodules; M (c) satisﬁes A.C.C. on submodules;

Proof :(d) is ﬁnitely generated.

Without proof. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

4 The Jacobson radical and Nakayama’s Lemma 10

The Jacobson radical is one of the most important two-sided ideals of a ring. As we will see in the next sections and Chapter 2, this ideal carries a lot of information about the structure of a ring and that of itsProposition-Deﬁnition modules. 4.1 (Annihilator / Jacobson radical)

M R R pMq “ tr P R | rm “ @ m P Mu R annihilator M (a) Let be an -module. Then ann : 0 is a two-sided ideal Jacobson radical R of , called of . ˆ (b) The JpRq “of is the two-sidedR pV q “ tx idealP R | ´ axb P R @ a; b P Ru : V R č : ann 1 simple -module ˝ V R I C R V – R {I R (c) If is a simple -module, then thereJp existsRq “ a maximalI: left ideal such that (as -modules) and ICR; I č

maximal left ideal Proof : Commutative Algebra

ExerciseSee 4.2 .

R R{JpRq

R{JpRq R (a) Prove that any simple -module may be seen as a simple -module. R ÝÑ R{JpRq (b) Conversely, prove that any simple -module may be seen as a simple -module. R R{JpRq [Hint: use a change of the base ring via the canonical morphism .]

Theorem(c) Deduce 4.3 (Nakayama’s that and Lemma) have the same simple modules. M R JpRqM “ M M “

ProofIf : is aCommutative ﬁnitely generated Algebra -module and , then 0.

RemarkSee 4.4 . M{U U R M One often needs to apply Nakayama’s Lemma to a ﬁnitely generated quotient module , where is an -submodule of . In thatM “ caseU ` theJpR resultqM mayùñ beU restated“ M as follows: Skript zur Vorlesung: Modular Representation Theory WS 2020/21

5 Indecomposability and the Krull-Schmidt Theorem 11

indecomposability

We now consider the notion of in more details. Our ﬁrst aim is to prove that inde- composabilityDeﬁnition 5.1 can be recognised at the endomorphism algebra of a module. R local ðñ RzRˆ R

ExampleA ring 1 is said to be : is a two-sided ideal of . K K zK ˆ “ t u

p R “ t a P Q | p bu RzRˆ “ t a P R | p|au (a) Any field is local because 0 byb definition.- b R (b) Exercise: Let be a prime number and : a a .::: Prove an that a ::: an´ and deduce thatK is local. R “ A “ 1 2 P MnpK q ¨ 0 1 1 ˛ ! ::: a ) ˆ (c) RExercise:zR “ tA LetP R | bea “ a fieldu and let : R ˝ . . . ‚ . Prove that 0. 0 .. .1 1 Proposition 5.2 0 and deduce that is local. R

R Let be a ring. Then TFAE: RzRˆ “ JpRq JpRq R (a) is local; R{JpRq (b) , i.e. is the unique maximal left ideal of ;

Proof :(c) N “ RiszR aˆ skew-ﬁeld.

ñ I C R ñ I Ď N Set : . (a) (b): Clear: proper left ideal Jp.R Hence,q “ by Proposition-DeﬁnitionI Ď N: 4.1(c), ICR; I č

maximal N R N left ideal N “ JpRq ñ Now,JpRq by (a) is an ideal of , hence mustR be a maximal leftR ideal,‰ evenR the{JpR uniqueq ‰ one. It follows thatpbq . r P RzJpRq “ R ˆ r ` JpRq P pR{JpRqqˆ R{JpRq (b) (c): If is the unique maximal left ideal of , then in particular 0 and 0. So let ñ R{JpRq R{JpRq ‰ R ‰ a P RzJpRq . Thena obviously` JpRq P pR{JpRqqˆ D b.P ItR followszJpRq that is a skew-field. (c) (a): Since is a skew-field by (c), 0, so that 0 and there exists . ab ` JpRq “ ` JpRq P R{JpRq Moreover, again by (c), , so that such that D c P JpRq ab “ ć R 1 D d P R abd “ p ´ cqd “ ñ a P R ˆ RzJpRq “ R ˆ ˆ Therefore,RzR “ JpRq such that 1 , which isR invertible in by Proposition-Definition 4.1(b). Hence such that 1 1 . Therefore , and it follows that which is a two-sided ideal of . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Proposition 5.3 (Fitting’s Lemma) 12

M R ' P R pMq M n P Zą

Let ben an -modulen`i which has a composition series and let End be an endomorphism ' pMq “ ' pMq 0 i ě of . Then there exists such that p'nq “ p'n`iq i ě (i) for every 1; M “ 'npMq ‘ p'nq : (ii) ker ker for every 1; and

Proof(iii) : ker M

By Corollary 3.4 the module satisﬁes'pM bothq Ě A.C.C.' pMq and Ě :::; D.C.C. on submodules. Hence the two chains of submodules p'q Ď 2 p' q Ď ::: 2 n ker ker M “ 'npMq ‘ p'nq eventually become stationary. Therefore we can ﬁnd an index satisfying both (i) and (ii). PropositionExercise: 5.4 Prove that ker . M R

Let be an -moduleM which has a compositionðñ series.R Then:pMq

is indecomposable End is a local ring. Proof : ñ M ' P R pMq n n n n P Zą M “ ' pMq ‘ p' q M ' pMq “ M n n n “ ”: pAssume' q “ that' pMisq indecomposable.“ p' q “ LetM End . Then by Fitting’s Lemma there exists 0 ¨ such that ' ker . As is indecomposable either and ker¨ 0 or '0 and ker . ˆ In theN ﬁrst“ caseR pMisqz bijective,R pM henceq “ t invertible. R pMqu In the second case is nilpotent. Claim: N R pMq Therefore, : End End nilpotentm elements of End . ' P N m P Zą ' “ is a two-sided ideal of End . 0 'm´ p'ρq “ “ pρ'q'm´ @ ρ P pMq : Let and minimal such that 0. Then R 1 1 'm´ ‰ 'ρ ρ' 'ρ; ρ' P N 0 End 1 '; ρ P N ' ` ρ “ ψ R pMq As 0ψ, ´ ρ;and ψ´ ' PcannotN be invertible, hence . Next let 1 . If 1 : were invertible in End , then by the previous argument we ψ´ ' “ ψ´ pψ ´ ρq “ ´ψ´ ρ would have , which would be nilpotent. HenceM 1 1 1

´ ´ ´ Id ´ ´ a´ ψ ρ ñ p M ´ψ ρqp M `ψ ρ ` pψ ρq ` ¨ ¨ ¨ ` pψ ρq q “ M ´ a awould be invertible.1 pψ ρq “ 1 1 1 2 1 1 (Indeed, nilpotent 1 Id ' ` ρ PIdN N Id , where is minimal such that 0.) R pMq This is a contradiction. Therefore , which proves that is an ideal. ð M M ;M M “ M ‘ M Finally, it follows from the Claim and the deﬁnition that End is local. 1 2 1 2 “ ”: Assume is decomposable and let be proper submodules such that . Then π M ‘ M ÝÑ M ‘ M ; pm ; m q ÞÑ pm ; q consider the two projections

1 1 2 1 2 1 2 1 : 0 Skript zur Vorlesung: Modular Representation Theory WS 2020/21

13 M M

1 2 π M ‘ M ÝÑ M ‘ M ; pm ; m q ÞÑ p ; m q onto along and M M 2π ; π1 P 2 pMq 1 π 2; π 1R 2 pMqˆ 2 : R R 0 π “ M ´π 2 1 1 2 1 2 onto along . Clearly End but π R EndJp R pMqqsince they are not surjective 2 1 by construction. Now, as Id is not invertible it follows from the characterisation of the pMqz pMqˆ ‰ 1J p pMqq Jacobson radical of Proposition-DeﬁnitionR 4.1R(b) that EndR . Therefore pMq End EndR End

and it follows from Proposition 5.2R that End is not a local ring.

Next, we want to be able to decompose -modules into direct sums of indecomposable submodules. The Krull-SchmidtProposition 5.5 Theorem will then provide us with certain uniqueness properties of such decompositions. M R M M R Let be an -module. If satisfies either A.C.C. or D.C.C., then admits a decomposition into Proofa :direct sum of finitelyM many indecomposable -submodules. M M “ M ‘ W Let us assume that is not expressible as a finite direct sum of indecomposableW submodules. Then 1 1 in particular Wis“ decomposable,M ‘ W soM that weW may write asW a direct sum of two proper 1 submodules.W W.l.o.g. we may assume that the statement is also false for . Then we also have a 1 2 2 2 2 1 decomposition , where and are proper sumbodules of with the statement being 2 W Ľ W Ľ W Ľ ¨ ¨ ¨ ; false for . Iterating this argument yields the following infinite chains of submodules: M Ĺ M ‘1 M Ĺ2 M ‘3M ‘ M Ĺ ¨ ¨ ¨ :

1 1 2 1 2 3

TheoremThe ﬁrst 5.6 chain (Krull–Schmidt contradicts D.C.C.) and the second chain contradicts A.C.C.. The claim follows. M R

1 1 1 1 Z Let be an -moduleM which“ M ‘ has ¨ ¨ ¨a ‘ compositionMn “ M ‘ series. ¨ ¨ ¨ ‘ M If n pn; n P ą q

M 1 1 0 R 1 1 n “ n π P Sn Mi – Mπpiq ď i ď n are two decomposition of into direct sums of ﬁnitely many indecomposable -submodules, then n , and there exists a1 permutation 1 such that for each 1 and M “ Mπp q ‘ ¨ ¨ ¨ ‘ Mπprq ‘ Mj ď r ď n: j“r` 1 à 1 for every 1 Proof : ď i ď n

π M “ M ‘ ¨ ¨ ¨ ‘ M Ñ M ; m ` ::: ` m ÞÑ m For each 1 let i n i n i i 1 1 ď j ď n1 : 1 1 1 1 1 1 ψ M “ M ‘ ¨ ¨ ¨ ‘ M 1 Ñ M ; m ` ::: ` m 1 ÞÑ m be the projection on the -thj factor of ﬁrst decomposition,n j and for each n1 j let

j 1 1 :

be the projection on the -th factor of second decomposition. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

14 Claim ψ P R pMq π ˝ ψ|M M Ñ M

M “ ψpM 1q ‘ M 1 ‘ ¨ ¨1 ¨ ‘ M 1 ψpM q – M : : if End is such that : n is an isomorphism, then Indeed 1 2 ψ| M Ñ ψpM q 1 π |1 ψpM q Ñ M M and ψpM q ψpM q X pπ q “ m P M m1 P ψpM q 1 1 1 1 1 1 1 1 1 π pmq “: Byπ p them q assumptionm ´ ofm theP claim,pπ q both : and : must be 1 1 1 isomorphisms. Therefore ker 0, and for every there exists 1 such that 1 1 1 M “ ψpM q `1 pπ q1 “ ψpM q ‘ pπ q “ ψpM q ‘ M ‘ ¨ ¨ ¨ ‘ M : , hence ker . It follows that n

1 1 1 1 1 2 ker ker n1 n1 M “ j“ ψj M “ j“ π ˝ ψj |M P R pM q M Hence the ClaimM holds. R pM q π ˝ ψj | P 1 1 M ř 1 ř 1 1 1 R pM q M Now, we have Id , and so Id End . But as has a composition1 1 1 j 1 series, so has , and therefore End is local by Proposition 5.41 . Thus if all the 1 End are not invertible, they are all nilpotent and then so is Id , which is in turn not invertible. π ˝ ψ | M Ñ M This is not possible, hence it follows that therej existsM an index such that

1 1 M “ ψj pM1 q ‘ M1 ‘ ¨ ¨ ¨ ‘ Mn ψj pM q – M 1 : 1 πp q “ j ψj pM q Ď Mj Mj 1 2 1 1 is an isomorphism and the Claim implies that and . 1 1 1 We then set 1 : . By deﬁnition ψj pM q – Masj “ Misπp indecomposbale,q : so that

1 1

n 1 1 Finally, an induction argument (Exercise!M “ Mπ)p yields:q ‘ ¨ ¨ ¨ ‘ Mπprq ‘ Mj ; j“r` 1 à 1 1 1 Mπpiq – Mi ď i ď r r “ n n “ n

mit (1 ). In particular, the case implies the equality . Chapter 2. The Structure of Semisimple Algebras

R R simple R In this chapter we study an important class of rings: the class of rings which are such that any - module can be expressed as a direct sum of -submodules. We study the structure of such rings through a series of results essentially due to Artin and Wedderburn. At the end of the chapter we will assume that the ring is a ﬁnite dimension algebra over a ﬁeld and start the study of its representation theory.Notation R

: throughout this chapter, unless otherwise speciﬁed, we let denote a unital and associative References:ring.

Methods of representation theory. Vol. I

[CR90] C. W. Curtis andGroup I. Reiner. representation theory. Part B: Modular representation. John Wiley theory & Sons, Inc., New York, 1990. [Dor72] L. Dornhoﬀ. Representations of ﬁnite groups . Marcel Dekker, Inc., New York, 1972. [NT89] H. Nagao andAdvanced Y. Tsushima. modern algebra. 2nd ed. . Academic Press, Inc., Boston, MA, 1989. [Rot10] J. J. Rotman. Providence, RI: American Mathematical Society (AMS), 2010. 6 Semisimplicity of rings and modules

ToProposition begin with, 6.1 we prove three equivalent characterisations for the notion of semisimplicity. M R

M M “‘iPI Si tSiuiPI R M If is an -module, then the following assertions are equivalent: M “ Si tSiuiPI R M (a) is semisimple,iPI i.e. for some family of simple -submodules of ; řR M Ď M M D R M Ď M (b) M “forM some‘ M family of simple -submodules of ; 1 2 (c) every -submodule admits a complement in , i.e. an -submodule 1 2 such that .

15 Skript zur Vorlesung: Modular Representation Theory WS 2020/21

16 Proof : ñ

ñ M “ iPI Si Si R M i P I M Ď M (a) (b): isR trivial. M J Ď I ř 1 (b) (c): Write , where is a simple -submodule of for each . Let be an

-submoduleiPJ Si of . Then consider the family, partially ordered by inclusion, of all subsets suchM thatX S “ ř iPJ i (1) is a direct sum, and 1 ř (2) 0. J Clearly this family is non-empty since it contains the empty set. Thus Zorn’s Lemma yields the M0 1 “ M ` S “ M ‘ S ; existence of a maximal element . Now, set i i iPJ iPJ 1 ÿ 1 ÿ : 0 0 M “ M1 1 1 Si Ď M i P I j P I Sj Ę M Sj 1 Swherej X M the“ second equality holds by (1) and (2). Therefore, it suﬃces to prove that , i.e. that for every . But if is such that , the simplicity of implies that 0 and it follows that 1 M ` Sj “ M ‘ Si ‘ Sj ˜iPJ ¸ 1 ÿ J 0

ñ M0 “ in contradiction with the maximality of . The claim follows. ñ M R 1 M M Ď M (b) (a): followsM from the argumentM “ aboveM ‘ M with M “0. M ‰ M 1 2 (c) (b): Let Rbe the sum of all simple -submodules in N. By (c) thereN existsĎ M a complement M 1 1 2 2 2 2 to , i.e. such thatM “ M. If“ M 0, we are done. If 0, then must contain a 1 1 simple -submodule (Exercise: prove this fact), say . But then by deﬁnition of , a 2 1 Examplecontradiction. 2 Thus 0 and so .

S ;:::;S R S ‘:::‘S (a) The zero modulen is completely reducible, but neither reducible nor irreducible!n 1 1 (b) If are simple -modules, then their direct sum is completely reducible by deﬁnition. a a K A K a | a ; a P K (c) The followingA exerciseV shows“ K that thereA exists modules which are not1 completely reducible. ` 1 ˘ 1 ( Exercise (Sheet 1): Let be a ﬁeld and let be the -algebra 0 . Con- tp x q | x P K u 2 A V sider the -module : , where acts by left matrix multiplication. Prove that: V (1) 0 is a simple -submodule of ; but (2) is not semisimple.

(d) Exercise (Sheet 1): Prove that any submodule and any quotient of a completely reducible module is again completely reducible. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Theorem-Deﬁnition 6.2 (Semisimple ring) 17 R semisimple

R A ring satisfying the following equivalent conditions is called . R (a) All short exact sequences of -modules split. R (b) All -modules are semisimple. R R˝ (c) All ﬁnitely generated -modules are semisimple.

(d) The regular left -module is semisimple, and is a direct sum of a finite number of minimal Proof : left ideals. ñ ˝ First, (a) and (b) are equivalent as a consequence of Lemma D.4 and the characterisationR “ of semisimpleiPI Li tLiuiPI modules given by Proposition 6.1(c). The implication (b) (c) is trivial, and it is also trivialÀ that (c) implies thei first; : : : ; claim in P I of (d), whichR in“ turnxi ` implies::: ` x thein secondxij P claimLij of (d). Indeed,ď j ď ifn for a P R some family of minimal left ideals.1 Then, by definition of a direct sum, there exists a finite number 1 of indices such that 1 with for each 1 . Therefore each a “ a ¨ “ ax ` ::: ` ax may be expressed in the form R i in R ˝ “ L ` ::: ` L 1 i in 1 ñ R M 1 and henceR . M “ mPM R ¨ m R ¨ m M R R ˝ R ¨ m Therefore, it remains to prove that (d)ř (b). So, assume that satisfies (d) and let be an arbitrary non-zero -module. Then write M . Now, each cyclic submodule of is isomorphic to an -submodule of , which is semisimple by (d). Thus is semisimple as well by Example2(d). ExampleFinally, 3 it follows from Proposition 6.1(b) that is semisimple. V K n K te ; ¨ ¨ ¨ ; enu V V “ K e ‘ ::: ‘ K en K pK eiq “ FieldsK are ei semisimple. Indeed,K if is a finite-dimensionalď i ď n vector space over a field of dimension , 1 1 then choosing a -basis of yields , where dim 1, hence is a simple -module for each 1 . Hence, the claim follows from Theorem- CorollaryDefinition 6.3 6.2(c). R

R˝ Let be a semisimple ring. Then: R (a) has a composition series;

Proof :(b) is both left Artinian and left Noetherian. R ˝ R ˝ (a) By Theorem-DefinitionR ˝ 6.2(d) the regular module admits a direct sum decomposition into a finite number of minimal left ideals.R Removing one ideal at a time, we obtain a composition series for . (b) Since has a composition series, it satisfies both D.C.C. and A.C.C. on submodules by Corol- lary 3.4. In other words, is both left Artinian and left Noetherian. J-semisimplicity Next, we show that semisimplicity is detected by the Jacobson radical. This leads us to introduce a slightly weaker concept: the notion of . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Deﬁnition 6.4 (J-semimplicity) 18 R J-semisimple JpRq “

Exercise 6.5 (Sheet 1) A ring is said to be if 0. R “ Z JpZq “ Z Z J Let . Prove that 0, but not all -modules are semisimple. In other words, is Proposition 6.6 -semisimple but not semisimple. R J

ProofAny : ñ left ArtinianR ring‰ is R-semisimple if and only if it is semisimple. I Ĳ R R ‰ I ‰ R I 0 “ R”: Assume 0 and is not semisimple. Pick a minimal left ideal (e.g. a minimal 0 0 elementClaim: I of the family of non-zeroR principal˝ left ideals of ). Then 0 since seen as an Indeed:-module is simple. 0 is a direct summand of .I ‰ “ JpRq “ I since ICR; 0 I č 0 m R maximal I I X m “ t u C left ideal R ˝ “ I ‘ m R{m 0 0 0 0 there exists a maximalm ‰ left ideal which does notI containm . Thus ‰ I ‰ m 0 andR so we 0 0 0 must have , as is simple.I HenceI the Claim. R ˝ m 0 1 0 1 0 Notice that then 0, and pick a minimalm left ideal min “ .I Then‘ m 0 , else would 1 1 0 be semisimple. The Claim applied to yields that is a direct summand of , hence also in . 1 0 1 1 Therefore, there exists a non-zero leftm idealĽ m Ľsuchm thatĽ ¨ ¨ ¨ . Iterating this process, we obtain an inﬁnite descending chain of ideals 0 1 2 ð R R ˝ – R{JpRq ‘ JpRq contradictingR D.C.C. “ ”: Conversely, if is semisimple,JpRq “ thenJpRq ¨ pR{JpRq ‘ JpRqq “ byJpR Theorem-Deﬁnitionq ¨ JpRq 6.2 and so as -modules, JpRq “

so that by Nakayama’s Lemma 0. Proposition 6.7 R{JpRq J

ProofThe : quotient ring is -semisimple.R R “ R{JpRq

Since by Exercise 4.2 the rings and : have the same simple modules (seen as abelian JpRq “ pV q “ pV q ` JpRq “ JpRq{JpRq “ groups), Proposition-Deﬁnition 4.1(a)R yields: R V V R č R č simple ann simple ann 0 -module -module

7 The Artin-Wedderburn structure theorem

The next step in analysing semisimple rings and modules is to sort simple modules into isomorphism classes. We aim at proving that each isomorphism type of simple modules actually occurs as a direct summand of the regular module. The ﬁrst key result in this direction is the following proposition: Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Proposition 7.1 19

M R tMiuiPI R M i P I Let be a semisimple -module. Let be a set of representatives of the isomorphism classes of simple -submodules of and for eachHi “ set V: V ĎM Vÿ–Mi :

M – H Then the followingiPI i statements hold: À R H M (i) ; i i 1 pH ;H 1 q “ t u i ‰ i (ii) everyR simplei i -submodule of is isomorphic to ; M “ V M (iii) Hom jPJ j 0 if ; and À (iv) if is an arbitraryH decompositioni “ Vj “ of intoVj “ aH directi : sum of simple submodules, then jPJ jPJ Vjÿ–Mi Vàj –Mi r :

Proof :

Claim 1: M “ jPJ Vj W R M D j P J WeW shall– Vj prove several statements which, taken together, will establish the theorem. À Iftπj M “ jPasJ V inj ÝÑ (iv) andVj ujPJ is an arbitrary simple -submodule of j , then such D j P J π pW q ‰ π | W ÝÑ V R W V that . j À j W j j Indeed: if : denote the canonical projections on the -th summand, then Claim 2: M “ V M “ H i P I R such that jPJ j 0. Hence : iPI isi an -isomorphism as both and are simple. Hi ÀMi À If as in (iv), then r and for each , every simple -submodule of r Hisi isomorphic to . ClaimIndeed: 3: theW 1st statement of the claimR is obvious andM the 2nd statement followsi P fromI Claim 1W appliedĎ H r i to . i P I W – Mi w P W zt u w “ jPJIfwj Pis anjPJ arbitraryVj w simplej P Vj -submodule of , then there is a unique such thatwj ‰ r . πIndeed:j pW řq ‰ it is clearÀ thatW there– V isj a unique wj “such that Vj – Mi . Now considerw P Hi 0 WandĎ writeHi with . The proof of Claim 1 shows that if any summand 0, then 1 Claim 4: R pHi; Hi1 q “ t u i ‰ i r r 0, and hence . Therefore 1 0 unless , and hence , so that . ‰ f P R pHi; Hi1 q i ‰ i R W Hi fpW qHom ‰ r r W 0 if .f|W W ÝÑ fpW q R Indeed:fpW ifq 0 HomR r r and Hi1 , then there mustMi exist a simple -submodule of r such that R 0, hence asHi1 is simple, : Mi1 ﬂ Mi is an -isomorphism. It follows from Claim 2, that is a simple -submodule of r isomorphic to . This contradicts Claim 2 saying that every Hi Ď Hi Hi Ď Hi simple -submodule of r is isomorphic to . Hi “ Hi i P I Now, it is clear that r by deﬁnition. On the other hand it follows from Claim 3, that r . r Hence for each , hencetH (iv).iuiP ThenI Claim 2 yields (i) and (ii), and Claim 4 yields (iii).

We give a name to the submodules deﬁned in Propostion 7.1: Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Deﬁnition 7.2 20

M R S R S-homogeneous component M SpMq R M S (a) If is a semisimple -module and is a simple -module, then the MpRq R of , denoted , is the sum of all simple -submodules of isomorphic to .

Exercise(b) 7.3We let(Sheet 2) denote a set of representatives of the isomorphism classes of simple -modules. R

I R idempotent R D e P R Let be a semisimple ring. Prove the following statements. e “ e I “ Re I1 I R˝ “ I ‘ I1 1 1 1 (a) Every non-zero“ e2` e left ideale P I of eisP generatedI by anI “ Re of , in other words such that and . (Hint: choose a complement for , so that and I R pI;R˝q write 1 with and . Prove that .) R R (b) If is a non-zero left ideal of , then every morphism in Hom is given by right e P R R pReq – peReq multiplication with an element of ˝ . f ÞÑ efpeqe R pR q – R f ÞÑopfp q (c) If is an idempotent, then End (the opposite ring) as rings via the map Re op e R R . In particular End via 1 . eRe (d) A left ideal generated by an idempotent of is minimal (i.e. simple as an -module) if R R R and only if is a division ring. (Hint: Use Schur’s Lemma.) R˝ (e) Every simple left -module is isomorphic to a minimal left ideal in , i.e. a simple - submodule of .

WeDeﬁnition recall that: 7.4 (Centre) centre pR; `; ¨q ZpRq “ ta P R | a ¨ x “ x ¨ a @ x P Ru

TheoremThe 7.5 (Wedderburnof a ring ) is : . R

S P MpRq SpR˝q ‰ |MpRq| ă 8 If is a semisimple ring, then the following assertions hold.

(a) If , then 0. Furthermore, . R˝ “ SpR˝q ; SPMpRq (b) We have à SpR˝q R SpR˝qT pR˝q “ S ‰ T P MpRq where each homogenous component is a two-sided ideal of and 0 if SpR˝q . R ZpRq (c) Each is a simple left Artinian ring, the identity element of which is an idempotent element of lying in . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

21 Proof : R R R ˝ S P MpRq SpR ˝q ‰ (a) By Exercise 7.3(e) every simple left -module is isomorphic to a minimal left ideal of , i.e. a ˝ simple submodule of . Hence if R ,“ then Vj 0. Now, by Theorem-Definition 6.2, the regular module admits a decomposition jPJ à Vj R ˝ R Vj j P J |MpRq| ă 8 SpR ˝q “ V into a direct sum of a finite number of minimalVj –S leftj ideals of , and by Claim 1 in the proof of Proposition 7.1 any simple submodule of is isomorphic to for some . Hence . RÀ˝ “ SpR ˝q : (b) Proposition 7.1(iv) also yields and Proposition 7.1(i) implies that SPMpRq à R L SpR ˝q x P T pR ˝q T P MpRq ˝ ˝ ˝ NextS notice‰ T that eachLx homogeneousĎ T pR q component is'x a leftR idealÝÑ ofR ; m, sinceÞÑ mx it is by definitionR a sum of ˝ leftR ideals. Now letLx “be' ax minimalpLq left ideal contained in , and let for aL ˝ ˝ ˝ withS ‰ T . Then Lx “ and becauseSpR qT pR q: “ isS anpR q-endomorphism of , then either is zero or it is again a minimal left ideal, isomorphic to . However, as , we have 0. Therefore 0, which implies that is also a right ˝ ˝ ideal, henceR “ two-sided.SPMpRq eS SpR q “ ReS eS SpR q SpR ˝q “ Re “ e R e SpR ˝q (c) Part (b) impliesř that the homogeneousS S components areS rings. Then, using Exercise 7.3(a), we may write 1 eS , where R with idempotent.a P R Since a “is aT two-sidedPMpRq aT a P T pR ˝q SpR ˝qT pR ˝q “ S ‰ T P MpRq e e “ δ e ideal, inT fact . It follows that is an identity elementS T for ST ř. T To see that is in the centreT of , consider an arbitrary element and write with . Since 0 if , we have . Thus, as is eSa “ eS aT “ eS eT aT “ eSeT aT an identity element for theT M-homogeneousR T M component,R weT M haveR Pÿp q Pÿp q Pÿp q “ eSaS

“ aSeS

“ aT eT eS “ p aT eT qeS “ p aT qeS “ aeS : T M R T M R T M R Pÿp q Pÿp q Pÿp q L ‰ SpR ˝q L R S L “ SpR ˝q ˝ Finally, ifSpR q0 is a two-sided ideal in , then must contain all the minimalR left ideals of isomorphic to as a consequence of Exercise 7.3(check it!). It follows that and Scholiumtherefore 7.6 is a simple ring. It is left Artinian, because it is semissimple as an -module.

R teS | S P MpRqu e P ZpRq S P MpRq If is aS semisimple ring, then there exists a set of idempotent elements such that e e “ δ e S;T P MpRq (i) S T STforS each ; “ e (ii) R SPMpRq forS all ; R “ ř Re Re (iii) 1 SPMpRq ;S S À (iv) , where each centralis a simple ring. orthogonal Idempotents satisfying Property (i) are called idempotents, and idempotents satisfying Prop- erty (ii) are called . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Remark 7.7 22 R R˝

Remember that if is a semisimple ring, then the regular modulenS admits a composition series. R˝ “ SpR˝q – S Therefore it follows from the Jordan-Hölder Theorem that SPMpRq SPMpRq i“ à à à nS P Zą 1

Theorem 7.8 (Artin-Wedderburn) 0 for uniquely determined integers . R

R – M pD q ; If is a semisimple ring, then, as a ring, nS S S M R Pźp q

DS “ R pSq

op where : End is a division ring. U “ U ‘ ¨ ¨ ¨ ‘ Ur r P Zą R pUq r ˆ r pBeforei; jq we proceed withR p theUj ;U proofiq of the theorem, first recallR that if we haveφ U a directÝÑ U sum decomposition 1 0 inc: φ proj: ( φ “), pφ thenij q ďi;jEndďr isφ isomorphicij Uj ÝÑ U toÝÑ the ringU ÝÑ of Ui -matrices in which the Rentry lies in Hom . This is because any -endomorphism : may be written as 1 a matrixR of components where : R , and when viewed in this way -endomorphisms compose in the manner of matrix multiplication. (Known from the GDM-lecture ifProofis : a field. The same holds over an arbitrary ring .) ˝ R pR q – R ˝ ˝ By Exercise 7.3(c), we have R pSpR q;T pR qq “op S fl T End as rings. On the other hand, since Hom 0 for (e.g. by Schur’s Lemma, or by pR ˝q – pSpR ˝qq Proposition 7.1), the above observationR yields R S M R Pźp q ˝ End End R pSpR qq – MnS p R pSqq – MnS p R pSq q DS “ R pSq R pSq op op op where End End End . Therefore, setting : End yields the result. For by Schur’s Lemma End is a division ring, hence so is the opposite ring.

8 Semisimple algebras and their simple modules

K K From now on weK leave the theory of modules over arbitrary rings and focus on finite-dimensional algebras over a field . Algebras are in particular rings,K and since -algebras and their modules are in particular -vector spaces, we may consider their dimensions to obtainA further information. In particular, weK immediately see that finite-dimensional -algebras are necessarily left Artinian rings. nS Furthermore, the structure theoremsA˝ “ of the previousSpA˝q –section tell usS that if is a semisimple algebra over a field , then SPMpAq SPMpAq i“ à à à nS 1 S A˝ A˝ where corresponds to the multiplicity of the isomorphism class of the simple module as a direct summand of in any given decomposition of into a finite direct sum of simple submodules. We shall Skript zur Vorlesung: Modular Representation Theory WS 2020/21

23 A A see that over an algebraically closed ﬁeld the number of simple -modules is detected by the centre Exerciseof and also 8.1 ( obtainSheet 2information) about the simple modules of algebras, which are not semisimple. A K K

ZpAq K A Let be an arbitrary -algebra over a commutative ring . K A ‰ K ÝÑ ZpAq; λ ÞÑ λ K (a) Prove that is a -subalgebra of . A

(b) Prove that if is a ﬁeld and 0, then 1 is an injective - A “ M pK q ZpAq “ K K homomorphism. n n MnpK q (c) Prove that if , then I , i.e. the -subalgebra of scalar matrices. (Hint: A ˆ K use the standard basis of .) Z A a a K : (d) Assume is the algebra of 2 2 upper-triangularp q “ a | matricesP over . Prove that 0 ` 0 ˘ (

WeTheorem obtain 8.2 the following Corollary to Wedderburn’s and Artin-Wedderburn’s Theorems: A K S P MpAq A Let be a semisimple ﬁnite-dimensional algebra over an algebraically closed ﬁeld , and let SpA˝q – M pK q pSpA˝qq “ n be a simplenS -module.K Then the followingS statements hold: pSq “ n 2 (a) K S and dim ; pAq “ pSq (b) dimK S;PMpAq K |MpAq| “ ř pZpAqq 2 (c) dim K dim ;

Proof :(d) dim .

K “ K ApSq – K DS DS “ ApSq – K “ K ˝ ˝ (a) Since , Schur’s Lemma implies that End R “ SpA.q Henceop theSpAop divisionq – M ringnS pK q in the ˝ statementK pSpA ofqq “ thenS Artin-Wedderburn Theorem is End . Hence Artin- ˝ WedderburnSpA q (and2 its proof) appliednS to theS case yields . Hence dim . n “ n ¨ pSq ùñ pSq “ n (b) Since is a direct sum ofS copiesS K of , (a) yields: K S 2 dim dim

A – SPMpAq MnS pK q (c) follows directly from (a) and (b). ś (d) Since by Artin-WedderburnZ andpAq (a) – we haveZpMnS pK qq “ K ,nS clearly; S M A S M A Pźp q Pźp q I K pK nS q “

where dim I 1. The claim follows. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Corollary 8.3 24 A K A K pZpA{JpAqqq Let be a finite-dimensional algebra over an algebraically closed field . Then the number of Proofsimple : -modules is equalA to dimA{JpAq . |MpAq| “ |MpA{JpAqq| A{JpAq J We have observed that and have the same simple modules (see Exercise 4.2), hence . Moreover, the quotient is -semisimple by Proposition 6.7, hence semisimple by Proposition 6.6 because finite-dimensional algebras are left Artinian rings. Therefore |MpAq| “ |MpA{JpAqq| “ ZpA{JpAqq : it follows from Theorem 8.2(d) that K ` ˘ dim Corollary 8.4 A K A A K Let be a finite-dimensional algebra over an algebraically closed field . If is commutative, then Proofany : simple -moduleA has -dimension 1A. A “ ZpAq

First assume that is semisimple.|MpAq| “ As KispA commutative,q “ K pSq ; . Hence parts (d) and (c) of SPMpAq Theorem 8.2 yield ÿ ě 2 dim dimloooomoooon K pSq “ S P MpAq 1 A A A{JpAq which forces dim 1 for each A.{JpAq Now, if is not semissimple, thenA{Jp againAq we use the factK that and have the same simple modules (that is seen as abelian groups). Because is semisimple and also commutative, the argument above tells us that all simple -modules have -dimension 1. The claim follows. K

Finally, we emphasise that in this section the assumption that the ﬁeld is algebraically closed is in general too strong and that it is possible to weaken this hypothesis so that Theorem 8.2, Corollary 8.3 K “ K ApSq – K and Corollary 8.4A still hold.S Indeed, if is algebraically closed, then Part (b) of Schur’s Lemma tells us that End for any simple -module . This is the crux of the proof of Theorem 8.2. The following terminology describesDeﬁnition this 8.5 situation. A K

A split pSq – K A S Let be a ﬁnite-dimensionalA -algebra. Then: K 1 K splitting ﬁeld for A K 1 K 1 b A (a) is called if End for every simple -module ; and K

(b) anA extension ﬁeldK of is called a A if the -algebra is split.

OfRemark course 8.6 if is split then itself is a splitting ﬁeld for . K A

In factA for a ﬁnite-dimensional -algebra , the following assertions are equivalent:

(a) is split; S MpAq A ÝÑ K pSq

(b) the product, for running through , of the structural homomorphisms End Skript zur Vorlesung: Modular Representation Theory WS 2020/21

a P A K S ÝÑ S; m ÞÑ am K 25

(mapping to the -linear mapA{JpAq – K) inducespSq : an isomorphism of -algebras S M A Pźp q End

This is a variation of the Artin-Wedderburn Theorem we have seen in the previous section. Chapter 3. Representation Theory of Finite Groups

G G npK q Representationnˆ theoryn of finite groups is originally concernedK with the ways of writingn a finite group as a group of matrices, that is using group homomorphisms from to the general linear group GL ofG invertibleÝÑ npK q -matrices with coefficients in a field for some positive integer . Thus, we shall first define representations of groups using this approach. Our aim is then to translate such homomorphisms GL into the language of module theory in order to be able to apply the theory we have developed so far. In particular, our first aim is to understand what the general theory of semisimple ringsNotation and the Artin-Wedderburn theorem bring to the theory of representationsG of finite groups. K KG : throughoutfree of this finite chapter, rank unless otherwise specified,K we let denote a finite group and be a commutativefinitely generated ring. Moreover,KG in order to simplify some arguments, we assume that all -modules considered are when regarded as -modules. (This implies, in particular, that they References:are as -modules.)

Local representation theory

[Alp86] J. L. Alperin. Representations and cohomology.. Vol. 11. I Cambridge Studies in Advanced Mathe- matics. Cambridge University Press, Cambridge, 1986. [Ben98] D. J. Benson. Methods of representation. Vol. theory. 30. Cambridge Vol. I Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1998. [CR90] C. W. Curtis andGroup I. Reiner. representation theory. Part B: Modular representation. John Wiley theory & Sons, Inc., New York, 1990. [Dor72] L. Dornhoff. Representations of groups . Marcel Dekker, Inc., New York, 1972. [LP10] K. Lux andA H. course Pahlings. in finite group representation. Vol. theory 124. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010. [Web16] P. Webb. . Vol. 161. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016. 9 Linear representations of finite groups

To begin with, we review elementary deﬁnitions and examples about representations of ﬁnite groups.

26 Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Deﬁnition 9.1 (K-representation, matrix representation) 27

K -representation G ρ G ÝÑ pV q V – K n n P Zě K pV q “ K pV q (a)A of is a group homomorphism : GL , where matrix0 representation G K X G ÝÑ pK q n P Z ( ) is a free -module of ﬁnite rank and GL : Aut . n ě n degree 0 (b)A of over is a group homomorphism : GL ( ).

In both casesK the integer isK called the of the representation. ordinary representation pK q “ pK q - |G| modular (c) representationIf is a ﬁeld, then a pK-representationq | |G| (resp. a matrix representation) is called an if char 0 (or more generally if char ), and it is called a Remark 9.2 if char .

B V Both concepts of a representation and of a matrix representation are closely related. Indeed, recall that every choice of a basis ofαB yieldsGLpV q a ÝÑ groupn isomorphismpK q;' ÞÑ p'qB

p'qB ': GL B K ρ G ÝÑ pV q B V G (where denotes the matrix of w.r.t. the basis ). Thus, a -representation : GL ρ αB together with the choice of a basisG of givespV q rise to a matrixnpK q representation of :

X GGLÝÑ npK qGL K ρ G ÝÑ pK nq Conversely, any matrix representation : GL gives rise to a -representation g ÞÑ ρpgq K n ÝÑ K n; v ÞÑ Xpgqv : GL V “ K n v K n Xpgqv : ,

namely we set , see as a column vector expressed in the standard basis of and Exampledenotes 4 the standard matrix multiplication.

ˆ (a) If is an arbitrary finite group,ρ thenG ÝÑ pK q – K g ÞÑ ρpgq “ K Ø K : GL K G the trivial representation: Id 1 G X G ¨ G ˆ X ÝÑ X V is a -representation of , called of . K tex | x P Xu (b) If is a finite -set, i.e. a finite set endowed with a left action : , and is a free -module with basisρX G ÝÑ, thenpV q g ÞÑ ρX pgq V ÝÑ V ; ex ÞÑ eg¨x : GL K G permutation: representation X

is a -representation of , called the associated with . Two particularly interesting examples are the following: Skript zur Vorlesung: Modular Representation Theory WS 2020/21

28 G “ Sn n ě n X “ t ; ; : : : ; nu ρX natural representation Sn (1) if X “ G ( 1) is the symmetric¨ G groupˆ X ÝÑ on Xletters and and 1 2 G then ρX “is calledρ regular representationof ; G (2) if and the left action : is just the multiplication in , then reg Deﬁnition 9.3 (Homomorphism: is called ofrepresentations, the equivalentof representations. ) ρ G ÝÑ pV q ρ G ÝÑ pV q K G V ;V K 1 2 1 2 Let 1 : GL and 2 : GL be two -representations of , where are K α V ÝÑ V ρ pgq ˝ α “ α ˝ ρ pgq g P G two non-zero free -modules of ﬁnite rank. homomorphism of representations G-homomorphism ρ ρ 1 2 (a)A -homomorphism : such that 2 1 for each is called a ρ pgq 1 2 (or aV V ) between and . 1 α α 1 ö 1 V V ρ pgq

2 2 2 α K isomorphism of representations G-isomorphism K ρ ρ equivalent similar (b) isomorphicIf, moreover, is a -isomorphism,ρ then„ ρ it is called an (or a ), and the -representations 1 and 2 are called (or , or 1 2 Remark 9.4 ). In this case we write .

„ (a) Equivalent representations have the same degree.

(b) Clearly is an equivalence relation.

(c) In consequence, it essentially suffices to study representations up to equivalence (as it es- Definitionsentially 9.5 (G-invariant suffices to subspace, study groups irreducibility up to isomorphism).) ρ G ÝÑ pV q K G

K W Ď V G-invariant Let : GL be a -representation of .

(a)A -submodule is calledρpgq W Ď Wif @g P G: ` ˘ w P W w “ ρpgg´ qpwq “ ρpgq ρpg´ qpwq P ρpgq W ρpgq W “ W (In fact in this1 case the reverse` 1 inclusion˘ holds` as˘ well, since` for˘ each we can write ρ irreducible , hence G .) K V reducible G (b) KThe representationď W isĎ calledV if it admits exactly two -invariant -submodules, namely 0 and itself; it is called if there exists a proper non-zero -invariant -submoduleV 0 . K G

Notice that itself and the zero -module 0 are always -invariant. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Deﬁnition 9.6 (Subrepresentation) 29 ρ G ÝÑ pV q K W Ď V G K

ρ G ÝÑ pW q If : GL is a -representationW and is a -invariant -submodule, then g ÞÑ ρW pgq “ ρpgq|W W ÝÑ W subrepresentation: ρ GL K G : : ρ G ÝÑ pV q irreducible ρ is called a of . (This is clearly again a -representation of .)

With this deﬁnition, it is clear that a representation : GL is if and only if does not possess any non-trivial proper subrepresentation. 10 The group algebra and its modules

K G modules modules K G group algebra We now want to be able to see -representations of a group as , and more precisely as Lemma-Deﬁnitionover a -algebra 10.1 (Group depending algebra on) the group , which is called the :

group ring KG K gPG λgg λg P K @g P G ř The is the ring whose elements are the -linear combinations with λgg,` and additionµgg “ andpλ multiplicationg ` µgqg are givenλ bygg ¨ µhh “ pλgµhqgh gPG gPG gPG gPG hPG g;hPG ÿ ÿ ÿ ` ÿ ˘ ` ÿ ˘ ÿ and KG K K G KG group algebra of G over K group ring respectively. Thus is a -algebra, which as a -module is free with basis . Hence we usually Proofcall : the KG K rather thanG simply . G K ¨ KG ˆ KG ÝÑ KG KG By deﬁnition is a free K-module with basis , and the multiplicationK in is extended by - bilinearity to the given multiplication : K. It is then straightforward that bears both the structures of a ring and of a -module. Finally, axiom (A3) of -algebras follows directly from the Remarkdeﬁnition 10.2 of the multiplication and the commutativity of .

¨ “ Clearly:KG G ¨ K KG |G| 1 1 ; ¨ KG G the -rank of is ; ¨ K KG is commutative if and only if is an abelian group; KG if is a field, or more generally (left) Artinian, then is a left Artinian ring, so that by Hopkins’ Theorem a -module is finitely generated if and only if it admits a composition G KG ÝÑ KG g ÞÑ g´ g P G series. left KG M 1 right KG ´ Also notice thatG since m ïsg a“ group,g ¨ them map definedKG by for each is an anti-automorphism. It follows that1 any -module may be regarded as a -module via the right -action : . Thus the sidedness of -modules is not usually an issue. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

30 KG K KG KG K As isK a -algebra, we may of course consider modules over and we recall that any -module is in particular a -module. Moreover, we adopt the following convention, which is automatically sat- isfied if is a field. Convention KG K : in the sequel all -modules considered are assumed to be free of finite rank when regarded as -modules. Proposition 10.3 K ρ G ÝÑ pV q G KG V

(a) Any -representation : ¨ GLG ˆ V ofÝÑgivesV rise to a -module structure on , where the external composition law is deﬁnedpg; vq by ÞÑ the mapg ¨ v “ ρpgqpvq : K KG : KG pV; `; ¨q K extended by -linearity to the whole of .

(b) Conversely, every ρV-moduleG ÝÑ pdeﬁnesV q a -representation g ÞÑ ρV pgq V ÝÑ V ; v ÞÑ ρV pgq “ g ¨ v : GL G : :

Proof : of the groupV K. ` pV; `q

(a) Since KGis a -module it is equipped with an internal addition such that is an abelian group.KG It is then straightforwardK to check that the given external composition law definedλ P K above vverifiesP V the -module axioms. (b)A -module is in particular a -moduleλv for“ the p λ scalarG q ¨ v multiplication : defined for all and all by PKG : 1 KG loomoon ρV pgq P pV q

ρ pg g q “ ρ pg q ˝ ρ pg q Moreover, it follows from the -moduleV axiomsV that V GL and also that

g ; g P G ρV 1 2 1 2

1 2 Examplefor 5 all , hence is a group homomorphism. trivial KG-module K KG G Via Proposition 10.3 the trivial representation (Example4(a)) corresponds to the so-called , that is the commutative¨ ringG ˆitselfK ÝÑ seenK as a -module via the -action pg; λq ÞÝÑ g ¨ λ “ λ : K KG :

extended by -linearity to the whole of . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Exercise 10.4 31

ρreg G KG KG˝ Prove that the regular representation of deﬁned in Exampale4(b)(2) corresponds to the Convention ˝ regular -module via Proposition 10.3. KG KG KG˝ : In the sequel, when no confusion is to be made, we drop the -notation to denote the regularLemma 10.5-module and simply write instead of . ρ G ÝÑ pV q ρ G ÝÑ pV q V – V KG 1 1 2 2 1 2 Two representations : GL and : GL are equivalent if and only if Proof : ρ „ ρ α V ÝÑ V K ρ pgq “ α ˝ ρ pgq ˝ α´ g P G as -modules. v P V g P G 1 1 2 1 2 2 1 If and : is a -isomorphism such that for each , g ¨ αpvq “ ρ p1gqpαpvqq “ αpρ pgqpvqq “ αpg ¨ vq ; then by Proposition 10.3 for every and every we have α KG 2 α V ÝÑ1V KG K g P G v P V 1 2 hence is a -isomorphism. Conversely, if : is a -isomorphism, then certainly it is a α ˝ ρ pgq ˝ α´ pvq “ αpρ pgqpα´ pvqq “ αpg ¨ α´ pvqq “ g ¨ αpα´ pvqq2 “ g ¨ v “ ρ pgqpvq ; -homomorphism and for each and by Proposition 10.3 for each we have 1 1 1 1 ρ pg1q “ α ˝ ρ pgq ˝ α´ 1 g P G 2

1 2 1 hence for each . Remark 10.6 (Dictionary) KG K More generally, through Proposition 10.3, wedictionary may transport terminology and properties from - modules to -representations and conversely. This letsK us build the following translation : KG

-Representations -Modules K G ÐÑ KG ÐÑ K -representationK of ÐÑ -module KG degree K ÐÑ -rank KG homomorphism of -representations ÐÑ KGhomomorphism of -modules equivalent -representations isomorphism of -modules ρV ‘ ρV ÐÑ KG V ‘ V subrepresentation ÐÑ -submodule“ KG 1 2 1 2 direct sum of representations ÐÑ direct sumKG of -modulesK irreducible representation G ÐÑ simple ( irreducible)KG KG-module the trivial representationK ÐÑ the trivial KG-module the regular representation of the regular -module completelyK reducible -representationG ÐÑ KGsemisimple -module (= completely reducible) ::::::every -representation of is is semisimple completely reducible Skript zur Vorlesung: Modular Representation Theory WS 2020/21

32 KG KG

FinallyProposition-Deﬁnition we introduce an 10.7 ideal (The of augmentationwhich encodes ideal a) lot of information about -modules.

ε KG ÝÑ K; gPG λgg ÞÑ gPG λg augmentation homomorphism (or map) pεq “ IpKGq augmentation ř ř idealThe mapKG: is an algebra homomorphism, called . Its kernel ker : is an ideal and it is called the IpKGq “ t λ g P KG | λ “ u “ pK q K IpKGq Ě JpKGq of . The followinggPG g statementsgPG hold:g KG KG{IpKGqř – K K ř (a) 0 ann and if is a ﬁeld ; IpKGq K |G| K tg ´ | g P Gzt uu (b) as -algebras;

Proof :(c) is a freeε KG-moduleÝÑ K of rank -1 with -basisK 1 1 ; ˆ G ÝÑ K Ď K ; g ÞÑ K KG Clearly,KG the map : is the unique extension by -linearity of the trivial representation 1 to , hence is an algebra homomorphism and its kernel is an ideal of the IpKGq “ pεq “ t λgg P KG | λg “ u ε algebra . gPG gPG pK q ř KG ř (a) ker 0 by definition of . The second equality is obvious by definition of ann , and the last inclusion follows from the definition of the Jacobson radical. gPG λgg P IpKGq gPG λg “ (b) follows from the 1st isomorphism theorem. ř ř (c) Let λgg “ . Thenλgg ´ “ λg0g and´ henceλg “ λgpg ´ q “ λgpg ´ q ; gPG gPG gPG gPG gPG g G ÿ ÿ ÿ ÿ ÿ Pÿzt u 0 1 1 tg ´ | g P Gzt uu IpKGq K1

which proves that the set 1 1 generates as a -module. The above computations also show that λgpg ´ q “ ùñ λgg “ g G gPG Pÿzt u ÿ 1 0 0 λg “ @ g P G 1 tg´ | g P Gzt uu K K IpKGq Hence 0 , which proves that the set 1 1 is also -linearly independent, Lemma 10.8hence a -basis of . K p G p IpKGq “ JpKGq

ExerciseIf is 10.9 a ﬁeld (Proof of positive of Lemma characteristic 10.8. Proceedand as indicated.is -group,) then . I R nil ideal I I R I (a) Recall that an ideal of a ring is called a if each element of is nilpotent. Accept g ´ g P Gzt u IpKGq the following result: if is a nil left ideal in a left Artinian ring then is nilpotent. KG (b) Prove that 1 is a nilpotent element for each 1 and deduce that is a nil IpKGq Ď JpKGq ideal of . IpKGq “ JpKGq (c) Deduce from (a) and (b) that using Exercise 3 on Exercise Sheet 2.

(d) Conclude that using Proposition-Deﬁnition 10.7. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

11 Semisimplicity and Maschke’s Theorem 33

Throughout this section, we assume that is a ﬁeld.

Our ﬁrst aim is to prove that the semisimplicity of the group algebra depends on both the characteristic ofTheorem the ﬁeld 11.1 and (Maschke the order) of the group. pK q - |G| KG K

' ψ ProofIf : char , then is a semisimple -algebra. L M N KG K By Theorem-Definition 6.2, we needK to prove that everyK s.e.s. 0 σ 0NofÝÑ M- modulesK splits. However, theψ field is clearly semisimple (again by Proposition-Definition 6.2). Hence any such sequence regarded as a s.e.s. of -vector spaces and -linear maps splits. So let : σ “ g´ σg N ÝÑ M be a -linear section for and|G| setgPG n ÞÑ g´ σpgnq 1 ř 1 |G| gPG r : : 1 1 ˆř |G| pK q - |G| |G| P K h P G. n P N

We may divide by σp,hn sinceq “ char g´ σpimpliesghnq “ thath pghq.´ Now,σpghn if q “ hσpandnq , then |G| |G| gPG gPG 1 ÿ 1 1 ÿ 1 r r ψ KG ψσpnq “ ψ g´ σpgnq “ g´ ψσpgnq “ g´ gn “ n ; |G| |G| |G| and gPG -lin gPG gPG 1 ÿ ` 1 ˘ 1 ÿ 1 1 ÿ 1 r ψσ “ N σ KG ψ

Examplewhere 6the last-but-one equality holds because Id . Thus r is a -linear section for . K “ C CG C pCq “

If is the ﬁeld of complex numbers, then is a semisimple -algebra, since char 0.

It turns out that the converse to Maschke’s theorem also holds, and follows from the properties of the augmentationTheorem 11.2 ideal. (Converse of Maschke’s Theorem) KG K pK q - |G|

ProofIf : is a semisimplepK q “ p -algebra, then charp | |G| . p KG Claim:Set char‰ V Ă :KGand letKG us assume that KG˝ . InV particularX IpKGq ‰must be a prime number. We have to prove that then is not semisimple. v “ gPG λgg P V zt u εpvq “ t “ hPG h If 0 is a -submodule of , then 0. ř ř Indeed: Let 0 . If εptq “0 we are“ done. |G| “ Else, set : . Then hPG ÿ pK q | |G| t P IpKGq 1 0tv tv P V V KG˝ tv P IpKGqzt u as char . Hence . Now consider the element . On the one hand since is a submoduletv “ h of λ,g andg “ on thep otherK ¨λg handqhg“ λg 0x “sinceεpvqx ñ εptvq“ εpvq “ |G|εpvq“ : hPG gPG h;gPG xPG gPG xPG xPG ´ ÿ ¯´ ÿ ¯ ÿ ÿ ´ ÿ ¯ ÿ ÿ 1 0 Skript zur Vorlesung: Modular Representation Theory WS 2020/21

34 IpKGq KG KG˝ KG˝ KG The Claim implies that , which is a -submodule by deﬁnition, cannot have a complement in . Therefore, by Proposition 6.1, is not semisimple and hence is not semisimple by Theorem- Deﬁnition 6.2. K KG

In the case in which the field is algebraically closed, or a splitting field for , the following exercise offers a second proof of the converse of Maschke’s Theorem exploiting the Artin-Wedderburn Theorem (TheoremExercise 11.38.2). (Proof of the Converse of Maschke’s Theorem for K splitting field for KG.) K p p | |G| KG

T “ x gPG gyK Assume is a ﬁeld of positive characteristic with and is a splitting ﬁeld for . Set ř KG KG˝ Ľ IpKGq Ě T Ľ : . KG˝ K (a) Prove that we have a series of -submodules given by 0. KG K (b) Deduce that has at least two composition factors isomorphic to the trivial module .

12 Simple modules over splitting ﬁelds

Assumption 12.1 K KG K is a splitting ﬁeld for G Throughout this section, we assume that is a splitting ﬁeld for , and we simply say thatK “ K . As explained at the end of Chapter 2 this assumption, slightly weaker than requiring that , implies that the conclusions of Theorem 8.2, Corollary 8.3 and Corollary 8.4 still hold. KG

We state here some elementary facts about simple -modules, which we obtain as consequences of theCorollary Artin-Wedderburn 12.2 structure theorem. K G KG If is a splitting ﬁeld for , then there are only ﬁnitely many isomorphism classes of simple Proof : -modules.

CorollaryThe 12.3 claim follows directly from Assumption 12.1 and Corollary 8.3. G K G KG

If is an abelian group and is a splitting ﬁeld for , then any simple -module is one- Proof : KG dimensional.

Since is commutative the claim follows directly from Assumption 12.1 and Corollary 8.4. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Corollary 12.4 35 p G p K G pK q “ p KG Let be a prime number. If is a -group, is a splitting field for and char , then the Prooftrivial : module is the uniqueJ simplepKGq “ IpKG-module,q upKG to{J isomorphism.pKGq – K K K ZpK q “ K By Lemma 10.8 we have . Thus as -algebras by Proposition- Definition 10.7(b). Now, as|MpKGis commutative,q| “ K ZpKG{JpKGqq, “ andK itK follows“ : from Assumption 12.1 and Corollary 8.3 that Remark 12.5 dim dim 1 |MpKGq| G K Another standard proof for Corollary 12.4 consists in using a result of Brauer’s stating that Corollaryequals 12.6 the number of conjugacy classes of of order not divisible by the characteristic of the field .

K G pK q - |G| |G| “ SPMpKGq K pSq ř 2 ProofIf : is a splittingpK q - ﬁeld|G| for and char KG , then dim .

Since char , the group algebra is semisimple by Maschke’s Theorem. Thus it follows from Assumption 12.1 and Theorem 8.2 that K pSq “ K pKGq “ |G| : S M KG P ÿp q 2 dim dim Chapter 4. Operations on Groups and Modules

In this chapter we show how to construct new -modules from old ones using standard module operations such as tensor products, Hom-functors, duality, or using subgroups or quotients of the initial group.Notation Moreover, we study how these constructions relate to each other.G K free of finite rank as K -modules: throughout this chapter, unless otherwise specified, we let denote a finite group and be a commutative ring. All modules over group algebras considered are assumed to be References: .

Local representation theory

[Alp86] J. L. Alperin. Representations and cohomology.. Vol. 11. I Cambridge Studies in Advanced Mathe- matics. Cambridge University Press, Cambridge, 1986. [Ben98] D. J. Benson. Methods of representation. Vol. theory. 30. Cambridge Vol. I Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1998. [CR90] C. W. Curtis and I. Reiner.Representations of groups . John Wiley & Sons, Inc., New York, 1990. [LP10] K. Lux andA H. course Pahlings. in ﬁnite group representation. Vol. theory 124. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010. [Web16] P. Webb. . Vol. 161. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016. 13 Tensors, ’s and duality

Deﬁnition 13.1 (Tensor product of KG-modules)

M N HomKG M bK N M N K KG diagonal action G If and are twoG -modules, then the tensor product of and balanced over becomes a -module via the of . In other words, the external composition law ¨ G ˆ pM b Nq ÝÑ M b N is deﬁned by the -action K K pg; m b nq ÞÑ g ¨ pm b nq “ gm b gn K : KG : extended by -linearity to the whole of .

36 Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Deﬁnition 13.2 (Homs) 37

M N KG K pM;Nq KG conjugation action G If Gand are two -modules, then the abelian group Hom becomes a -module via the so-called of . In other words, the external composition law is deﬁned by ¨ G ˆ K pM;Nq ÝÑ K pM;Nq the -action pg; fq ÞÑ g ¨ f M ÝÑ N; m ÞÑ pg ¨ fqpmq “ g ¨ fpg´ ¨ mq : Hom Hom K KG 1 : : N “ K KG K M˚ “ pM;K q extended by -linearity to the whole of . K

DeﬁnitionSpecifying 13.3 Deﬁnition (Dual 13.2 of a toKG-moduleyields) a -module structure on the -dual Hom .

M KG K M˚ KG

(a) If is a -module, then its -dual becomes a -module via the external composition ˚ ˚ law is deﬁned¨ byG theˆ mapM ÝÑ M pg; fq ÞÑ g ¨ f M ÝÑ K ; m ÞÑ pg ¨ fqpmq “ fpg´ ¨ mq

: 1 K : KG : M;N KG KG ρ P pM;Nq KG extended by -linearity to the whole of . KG

(b) If are -modules, then every -homomorphism Hom induces a - ˚ ˚ ˚ homomorphismρ N ÝÑ M f ÞÑ ρ˚pfq M ÝÑ K ; m ÞÑ ρ˚pfqpmq “ f ˝ ρpmq : : : :

Lemma 13.4(See Propotion D.3.) ˚ M N KG K pM;Nq – M bK N KG

ProofIf : and are -modules, then Hom as -modules. θ “ θ M˚ b N ÝÑ pM;Nq The map M;N K K f b n ÞÑ θpf b nq M ÝÑ N; m ÞÑ θpf b nqpmq “ fpmqn : : Hom K : g P G f P M˚ n P N m P M deﬁnes a -isomorphism. (Check it!) θpg ¨ pf b nqqpmq “ θpg ¨ f b g ¨ nqqpmq “ pg ¨ fqpmqg ¨ n Now, for every , , and , we have on the one hand “ fpg´ ¨ mqg ¨ n

g ¨ θpf b nq pmq “ g ¨ θpf b nqpg´ mq “ g ¨ fpg´ mqn “ fpg´ ¨ mqg ¨ n ; and on the other hand 1 1 1 θpg ¨ p`f b nqq “ g˘¨ θpf b nq` ˘ θ ` KG˘ ` ˘ hence and it follows that is in fact a -isomorphism. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Remark 13.5 38 ˚ M “ N KG K pMq – M bK M ˚ ˚ K pMq “ n tm ; : : : ; mnu K M tm ; : : : ; mnu K n ˚ ˚ In caseM P KthepMq above constructions yield a “-modulei“ mi structureb mi P M on bEndK M . 1 Moreover, if rk : , KG is a -basis of and 1 is the dual -basis, ř then Id End corresponds to the element r : 1 . (Exercise!) I K ÝÑ M˚ b M This allows us to deﬁne the -homomorphism: K ÞÑ : 1 r Deﬁnition 13.6 (Trace map) M KG trace map M KG

M˚ b M ÝÑ K If is a -module, then the M associatedK to is the -homomorphism f b m ÞÑ fpmq : Tr : Notation 13.7 M N KG M | N M N If and are -modules, we shall write to mean that is isomorphic to a direct Lemma 13.8 summand of . ˆ ˚ K pMq P K K | M bK M

Proof : I KG M If rk , then . K pMq ˚ ˚ M bK M – p M q ‘ K K | M bK M λ P1K rk By Lemma-Definition D.4(c) it suffices to check that is a -sectionn for Tr , because then ker Tr , hence . So let .λ Then ˚ M ˝ I pλq “ M pλrq “ M p mi b miq K pMq K pMq K pMq ” ı i“ 1 1 n ÿ λ Tr Tr Tr ˚1 rk rk “ rk m pmiq pMq i K i“ ÿ λ n “ rk 1 “ λ : pMq K i“ ÿ 1 M ˝ I “ K 1 K pMq rk 1 rk ExerciseHence 13.9 Tr Id . K M N KG M – pM˚q˚ KG Assume is a field (to simplify) and let , be -modules. Prove the following assertions: ˚ ˚ ˚ ˚ ˚ ˚ M ‘ N – pM ‘ Nq M bK N – pM bK Nq KG (a) as -modules (in a natural way); M M˚ (b) and as -modules (in a natural way);

(c) is simple, resp. indecomposable, resp.´ semisimple, if and only if is simple, resp. M KG M ˝ θ indecomposable, resp. semisimple; M;M ˚ 1 ˚ M | M bK M bK M | K pMq M ‘ M | M bK M bK M (d) Tr is a -homomorohism and Tr coincides with the ordinary trace of matrices; (e) , and if char(K) dim , then . (This is more challenging!) Skript zur Vorlesung: Modular Representation Theory WS 2020/21

K M 39 N K Can (a) to (e) be generalised to the case in which is an arbitrary commutative ring and and free of ﬁnite rank when seen as -modules ? 14 Fixed and coﬁxed points

K KG Fixed and cofixed points explain why in the previous section we considered tensor products and Hom’s overDefinitionand 14.1 not over (G-fixed. points and G-cofixed points) M KG

G-fixed points M MG “ m P M | g ¨ m “ m @ g P G Let be a -module. G-cofixed points M M “ M{pIpKGq ¨ Mq ( (a) The of are by definition :G .

(b) The of are by deﬁnition : .

¨ MG KG M G In other words: ¨ M M G G is the largest -submodule of on which acts trivially, and Lemma 14.2 is the largest quotient of on which acts trivially. G M;N KG K pM;Nq “ KGpM;Nq pM bK NqG – M bKG N

ProofIf : K are -modules,f M ÝÑ thenN Hom KGHom and fpg ¨ mq “ g ¨ fpmq . g P G m P M g´ ¨ fpg ¨ mq “ fpmq g P G m P M G A -linear map : f is a morphism1 of -modulesG if andK onlypM;N if q “ KGpM;Nqfor all and all , that is if and only if for all and all . This is exactly the condition that is ﬁxed under the action of . Hence Hom Hom . ExerciseSecond 14.3 claim: similar, Exercise! ' ψ K ÝÑ L ÝÑ M ÝÑ N ÝÑ KG M – L ‘ N

Let be a field and let 0 KG p0N;be´q a s.e.s. of -modules. Prove that if , then the s.e.s. splits. [Hint: Consider the exact sequence induced by the functor Hom (as in Proposition D.3(a)) and use the fact that the modules considered are all finite-dimensional.] 15 Inflation, restriction and induction

In this section we deﬁne new module structures from known ones for subgroups, overgroups and quotients,Remark and 15.1 investigate how these relate to each other. H ď G H ÝÑ G; h ÞÑ h K

{ KH ÝÑ KG; hPH λhh ÞÑ hPH λhh KH (a) If K is a subgroup,KG then the inclusion can be extended by -linearity ř ř to an injective algebra homomorphism : . Hence is a -subalgebra of . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

40 U E G G ÝÑ G{U g ÞÑ gU K π KG ÝÑ K rG{Us (b) Similarly, if is a normal subgroup, then the quotient homomorphism , can be extended by -linearity to an algebra homomorphism : .

It is clearinflation that we canrestriction always perform changes of the base ring using the above homomorphism in order to obtain new module structures. This yields two natural operations on modules over group algebras Definitioncalled 15.2and (Restriction) . H ď G M KG M KH G G { KH ÝÑ KG H pMq M ÓH Let restrictionbe a subgroup.M If Gis aH -module, then may be regarded as a -module through a change of the base ring along : , which we denote by Res or simply and Definition 15.3 (Inflation) call the of from to . U E G M K rG{Us M KG π KG ÝÑ K rG{Us G LetG{U pMq be a normalinflation subgroup.M If isG{ aU G -module, then may be regarded as a -module through a change of the base ring along : , which we denote by RemarkInf 15.4 and call the of from to . H ď G M KG ρ G ÝÑ GLpMq G K K M ÓH (a) If is a subgroup, is a -module and : is the associated { ρ -representation, then the -representationH ÝÑ G ÝÑ associatedGLpMq : to is simply the composite morphism U E G M K rG{Us ρ G{U ÝÑ GLpMq G K K G{U pMq (b) Similarly, if is a normal subgroup, is a -module and : π ρ is the associated -representation,G ÝÑ thenG{ theU ÝÑ-representationGLpMq : associated to Inf is simply Lemma 15.5

G G G H ď G M ;M KG pM ‘ M q ÓH “ M ÓH ‘ M ÓH U E G G G G M ;M K rG{Us G{U pM ‘ M q “ G{U pM q ‘ G{U pM q 1 2 1 2 1 2 (a) If and are two -modules, then . If 1 2 1 2 1G H G 2 (Transitivityand are of restriction) two -modules,L ď H ď G then InfM KG Inf M ÓHÓL “InfM ÓL . ˚ G G ˚ H ď G M KG pM q ÓH – pM ÓH q U E G M K rG{Us (b) G ˚ If G ˚ and is a -module, then . G{U pM q – p G{U Mq (c) If and is a -module, then . If and is a - module, then Inf Inf . Proof :

(a) Straightforward{L;H L ÝÑ H from the fact that the external compositionL H { lawH;G onH aÝÑ directG sum is deﬁned compo- nentwise.H G {L;G L ÝÑ G L G (b) If : denotes the canonical inclusion of in , : the canonical inclusion { ˝ { “ { : of in and : the canonicalH;G inclusionL;H ofL;G in , then Skript zur Vorlesung: Modular Representation Theory WS 2020/21

41 {L;G G H G {H;G {L;H M ÓHÓL “ M ÓL Thus performing a change of the base ring via is the same as performing two successive changes of the base ring via ﬁrst and then . Hence . (c) Straightforward.

ADeﬁnition third natural 15.6 operation (Induction comes) from extending scalars from a subgroup to the initial group. H ď G M KH KG pKG; K Hq induction M H G KG

Let be a subgroup and let beG a -module. Regarding as a -bimodule, we deﬁne the of from toH pMtoq be“ theKG leftbKH M:-module M G G M ÒH IndH p q : Example 7 We shall also write instead of Ind .

G H “ t u M “ K K Òt u“ KG bK K – KG

G H G (a) TransitivityIf 1 and of induction, then L1 ď H ď G M . KL M ÒL “ pM ÒL qÒH

(b) : clearly and is a -module, then by the associativity of the tensor product. H

RemarkFirst, we 15.7 analyse the structure of an induced module in terms of the left cosets of . G{H “ tgH | g P Gu H G rG{Hs rG{Hs “ tg ; : : : ; g|G H|u Recall that g “ denotes theg ; set : : : ; of g|G leftH| cosetsP G of in giH. Moreover,‰ gj H i we‰ j write 1 G for a set of representatives of theseH left cosets. In other words, : 1 1 (where we assume that 1) for elements : such that if and is the disjoint union of theG left“ cosets ofgH ,“ sog thatH \ ::: \ g|G H|H: gPrG{Hs ğ 1 :

KG “ gK H ; gPrG{Hs It follows that à gK H “ tg hPH λhh | λh P K @ h P Hu gK H – KH right KH gh ÞÑ h h P H ř where . Clearly, as -modules via |G H| for each . ThereforeKG – KH “ pKHq gPrG{Hs à : right KH KH rG{Hs and hence is a freeM -moduleKH with a -basis given by the left coset representatives in . In consequence,KG bKH ifM “is p a givengK H-module,q bKH M then“ we havepgK H bKH Mq “ pg b Mq ; gPrG{Hs gPrG{Hs gPrG{Hs à à à Skript zur Vorlesung: Modular Representation Theory WS 2020/21

g b M “ tg b m | m P Mu Ď KG bKH M: where we set g b M M K K : Clearly, each is isomorphic tog b MasÝÑ a M;-module g b m viaÞÑ m the : -isomorphism

G K p H pMqq “ |G H| ¨ K pMq : It follows that KG bKH M G K rk Ind : rk x P G xgi “ gj h h P H Next we see that with its left action on , the group permutes these -submodules: for if , then x ¨for pgi someb mq “ xg,i b andm hence“ gj h b m “ gj b hm : ď i; j ď |G H|

´ This action is also clearly transitiveg sincej gi p forgi b everyMq “1 gj b M: : we can write 1 g b M H g “ ´ gi b M giHgi 1 1 Exercise: Check that1 the stabiliser of is (where 1) and deduce that the stabiliser of is . Proposition 15.8 (Universal property of the induction)

H ď G M KH j M ÝÑ KG bKH M; m ÞÑ b m KH KG N KH G Let , let' MbeÝÑ a -moduleH pNq and let : KG 1 ' KGbeb theKH M canonicalÝÑ N map (which' ˝ isj in“ ' fact a -homomorphism). Then, for every -module and for every - homomorphism : Res , there exists a unique' -homomorphism ˜ : such that ˜ , or in other words suchM that the followingN diagram commutes: ö j D ' G pMq H ! ˜

Ind Proof :

The universal property of the tensor product yields the existence of a well-deﬁned homomorphism of ' KG b M ÝÑ N abelian groups KH a b m ÞÑ a ¨ 'pmq ˜ : KG m P M '˝jpmq “ 'p bmq “ ¨'pmq “ 'pmq . ' ˝ j “ ' a P KG m P M which is obviously -linear. Moreover, for each , we have ˜ ˜ 1 1 , 'pa b mq “ 'pa ¨ p b mqq “ a ¨ 'p b mq “ a ¨ p' ˝ jpmqq “ a ¨ 'pmq hence ˜ . Finally the uniqueness follows from the fact for each and each , we have ' ˜ ˜ 1 ˜ 1 ˜

hence there is a unique possible deﬁnition for ˜.

Induced modules can be hard to understand from ﬁrst principles, so we now develop some formalism that will enable us to compute with them more easily. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

43 G H coinduction KH M coinduction M H G To beginKG with, there is, in fact, a further operation that relates the modules over a group and a G subgroup called . Given a H pM-moduleq “ KH, thenpKG; the Mq of from to is the left -moduleKG KG KG Coind : Hom ¨ KG ˆ KH pKG; Mq ÝÑ KH pKG; Mq where the left -modulepg; structure θq is deﬁned ÞÑ g through¨ θ KG theÝÑ naturalM; x ÞÑ right pg ¨ θqp-modulexq “ θp structurex ¨ gq of : Example 8 : Hom Hom : : G ˚ ˚ H “ t u M “ K t upK q – pKGq KG pKGq

If 1 and KG, then Coind 1 (i.e. with theK -modulet u structureG p ofKGq˚ of Deﬁnition 13.3). Exercise: exhibit a -isomorphism between the coinduction of from 1 to and .

Now, we see that the operation of coinduction in the context of group algebras is just a disguised versionLemma of15.9 the (Induction induction andfunctor. coinduction are the same)

H ď G M KH KGbKH M – KH pKG; Mq KG KG – pKGq˚ KG If is a subgroup and is a -module, then Hom as -modules. ProofIn : particular, KG as -modules.

KG bKH M ÝÑ KH pKG; Mq Mutually inverse -isomorphismsg b m areÞÑ deﬁnedpg byb mq KG ÝÑ M; x ÞÑ pxgqm Φ : Hom Φ : KH pKG; Mq ÝÑ KG bKH M and ´ θ ÞÑ gPrG{Hs g b θpg q : Ψ : Hom 1 H “ t u M “ K ř

˚ KG – KG bK K – K pKG; K q – pKGq It follows that in the case in which 1 and , KG Hom as -modules. Here we emphasise that the last isomorphism isn’t an equality. See the previous Theoremexample. 15.10 (Adjunction / Frobenius reciprocity / Nakayama relations) H ď G N KG M KH K Let be a subgroup. Let be a -module and let be a -module. Then, there are KH pM; KGpKG; Nqq – KGpKG bKH M;Nq -isomorphisms: G G KH pM;N ÓH q – KGpM ÒH ;Nq (a) Hom Hom Hom , pN ÓG ;Mq – pN;M ÒG q or inKH other words,H Hom KG H Hom ;

Proof :(b) Hom Hom .

KGpKG; Nq – KG bKG N – N KG G KGpKG; Nq – N ÓH KH (a) Since induction and coinductionK coincide, we have Hom as - modules. Therefore, Hom as -modules, and it suﬃces to prove the second isomorphism. In fact, this -isomorphism is given by the map Skript zur Vorlesung: Modular Representation Theory WS 2020/21

G G 44 KH pM;N ÓH q ÝÑ KGpM ÒH ;Nq ' ÞÑ ' ' KG Φ : Hom ' Hom ' KG ' ˝ j “ ' ˜ where ˜ is the -homomorphism inducedG by by the universal propertyG of the induction. Since KGpM ÒH ;Nq ÝÑ KH pM;N ÓH q ˜ is the unique -homomorphism suchψ that ˜ ÞÑ ,ψ setting˝ j Ψ : Hom Hom K provides us with an inverse map forexteriorΦ. Finally, tracemap it is straightforward to check that both Φ and Ψ

are -linear.G G G H KH pN ÓH ;Mq ÝÑ KGpN;M ÒH q (b) Exercise: check that the so-called G G ´ ' ÞÑ H p'q N ÝÑ M ÒH ; n ÞÑ gPrG{Hs g b 'pg nq Trp : Hom Hom K ř 1 Trp :

Propositionprovides 15.11 us with the required -isomorphism. H ď G N KG M KH KG Let be a subgroup. Let be a -module and let be a -module. Then, there are pM b N ÓG qÒG – M ÒG b N -isomorphisms:K H H H K pM;N ÓG qÒG – pM ÒG ;Nq (a) K H H K ; andH

(b) Hom Hom . Proof :

pM b N ÓG qÒG “ KG b pM b N ÓG q – pKG b Mq b N “ M ÒG b N (a) It follows fromK theH associativityH KH of the tensorK H product thatKH K H K H ˚ G G ˚ M KH pM qÒH – pM ÒH q (b) WeKG push back the proof until we have introduced the concept of an -free module. (We will then prove that if is a -module, then and (b) will follow directly from (a) and Exercisethe 15.12-isomorphism of Lemma 13.4.) K U;V;W KG KG

pU b V;W q – pU;V ˚ b W q Let be aK ﬁeld.K Let be K -modules.K Prove that there are isomorphisms of -modules: pU b V;W q – pU;V ˚ b W q – pU; pV;W qq (i) HomKG K Hom KG K ; and KG K

(ii) Hom Hom Hom Hom . Chapter 5. The Mackey Formula and Cliﬀord Theory

Mackey decomposition formula The results in this chapter go more deeply into the theory. We start with the so-called , which provides us with yet another relationship between induction and restriction. After that we explain Clifford’s theorem, which explains what happens when a simple representation is restricted to a normal subgroup. These results are essential and have many consequences throughout representationNotation theory of finite groups. G K free of finite rank as K -modules: throughout this chapter, unless otherwisefinitely specified, generated we letKGdenote a finite group and be a commutative ring. All modules over group algebras considered are assumed to be References: (hence, in particular, they are as -modules).

Local representation theory

[Alp86] J. L. Alperin. Representations and cohomology.. Vol. 11. I Cambridge Studies in Advanced Mathe- matics. Cambridge University Press, Cambridge, 1986. [Ben98] D. J. Benson. Methods of representation. Vol. theory. 30. Cambridge Vol. I Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1998. [CR90] C. W. Curtis and I. Reiner.Representations of groups . John Wiley & Sons, Inc., New York, 1990. [LP10] K. Lux andA H. course Pahlings. in ﬁnite group representation. Vol. theory 124. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010. [Web16] P. Webb. . Vol. 161. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016. 16 Double cosets Deﬁnition 16.1 (Double cosets) H L G g P G

Given subgroups and of weHgL deﬁne“ t forhgk eachP G | h P H; k P Lu

G pH;Lq-double: coset g HzG{L pH;Lq G and call this subset of the of . Moreover, we let denote the set of -double cosets of .

45 Skript zur Vorlesung: Modular Representation Theory WS 2020/21

46 pH;Lq G

First,Lemma we 16.2 want to prove that the -double cosets partition the group . H;L ď G

pH;Lq H Let . L (a) Each -double coset is a disjoint union of right cosets of and a disjoint union of left pH;Lq rHzG{Ls cosets of . pH;Lq G (b) Any two -double cosets either coincide or are disjoint. Hence, letting denote a set of representatives for the G-double“ cosetsHgL of : , we have g H G L Pr ğz { s

Proof : hgk P HgL k P L hgk ¨ k “ hgpkk q P HgL L hgk HgL HgL L 1 1 1 (a) If and , thenHgL . ItH follows that the entire left coset of thatg ; contains g P G h isg containedk “ h g ink P Hg. ThisL X Hg provesL that g “ish´ ah uniong k ofk´ leftP cosetsHg L of .A HgsimilarL Ď argumentHg L proves thatHg L Ď Hgis aL union of right cosets of . 1 1 1 2 1 1 1 2 2 2 1 2 1 2 2 2 2 (b) Let . If , then 1 1 so that 1 2 2 1 . Similarly . Thus if two double cosets are not disjoint, they coincide. X G GzX G X rGzXs Y G Y {G rY {Gs If is a left -set we use the standard notation for the set of orbits of on , and denote a setX of representativesG for theses orbitsx P X by X –.G Similarly{ Gpx ifq is a right -set we write and . xWe shallG also repeatedly use theG orbit-stabiliser theorem without further mention: in other words, if is a transitive left -set and then Stab (i.e. the set of left cosets of the stabiliser of Exercisein ), and 16.3 similarly for right -sets.

H;L ď G pH;Lq HzpG{Lq pHzGq{L (a) Let . Prove that the set of -double cosets is in bijection with the set of orbits , and also with the set ofHgL orbitsÞÑ HpgLq P HunderzpG{ theLq mappings HgL ÞÑ pHgqL P pHzGq{L: HzG{L pH;Lq

G “ S H “ L “ S “ t ; p qu S This justiﬁes the notation for the set of -double cosets. 3 2 3 (b) Let . Consider : rS zSId{S 1s 2“ t as; p a subgroupqu of . Prove that

2 3 2 Id 1 2 3 S zS {S “ t t ; p qu; tp q; p q; p q; p quu : while 2 3 2 Id 1 2 1 2 3 1 3 2 1 3 2 3 Skript zur Vorlesung: Modular Representation Theory WS 2020/21

17 The Mackey formula 47

H L G KL L G H If and are subgroups of , we wish to describe what happens if we induce a -module from to Remarkand then 17.1 restrict it to . KG pKH;KLq

G G “ gPrHzG{Ls HgL We need to examine seen as a -bimodule (i.e. with left and right external laws by Ů multiplication in ). Since KG “ , weK havexHgLy gPrHzG{Ls à pKH;KLq K xHgLy K K HgL M KL gM g b M K p gLq

as -bimodule, where ´denotes the free -module with -basis . Now if is a -module, we willp alsogkg writeq ¨ pg b formq “ g b,km which is a left -module with 1 k P L m P M

for each and each K. WithxHgL thisy – notation,KH bK pH weX gL haveq pg b KLq ; hgk P HgL h b g b k

Theoremwhere 17.2 (Mackeycorresponds formula) to . H;L ď G M KL KH

G G g gL H Let and let be a M Ò-module.LÓH – Then, asp M ÓH-modules,X gLqÒHX gL : gPrHzG{Ls à

Proof : KH

G G G M ÒLÓH – pKG bKL MqÓH – K xHgLy bKL M It follows from Remark 17.1 that as left gPr-modulesHzG{Ls we have à – KH bK pHX gLq pg b KLq bKL M gPrHzG{Ls à gL – KH bK pHX gLq pg b MqÓHX gL gPrHzG{Ls à g gL H – p M ÓHX gLqÒHX gL : gPrHzG{Ls à

Remark 17.3

Z KZ mod KZ K Given an arbitrary ﬁnite group , write KLmodfor theKH categorymod of -modules which are free of ﬁnite rank as -modules. Then,G expressedG in categoricalH terms,gL the Mackey formula says that we H ˝ L “ HX gL ˝ HX gL ˝ pgq have the following equality of functors fromgPrHzG{Ls to : à pgq Res Indg P G Ind Res Inn

where Inn is conjugation by . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Exercise 17.4 48 H;L ď G M KL N KH

Let G , let G be a -moduleg g andL let beH a G -module. Use the Mackey formula to M Ò b N Ò – p M Ó g b N Ó g qÒ g prove that:L K H gPrHzG{Ls HX L K HX L HX L G ÀG g gL H G (a) K pM ÒL ;N ÒH q – gPrHzG{Lsp K p M ÓHX gL;N ÓHX;gLqqÒHX gL À (b) Hom Hom .

18 Cliﬀord theory

Cliﬀord’s theorem

We now turn to , which we present in a weak and a strong form. Clifford theory is a K collection of results about induction and restriction of simple modules from/to normal subgroups. S KG Throughout this section, we assume that is a field. K I K S I First we emphasiseK again, thatK {I this is no lossS of generality:pK {IqG indeed if were a simple -module with an arbitrary commutative ring, then letting be the annihilator in of , we have that is a maximalTheorem ideal 18.1 of (Clifford’s, so that Theorem,is a weak field form and ) is a -module. G U E G S KG S ÓU

G ProofIf : V is a normal subgroupKU and isS aÓ simpleU -module, then g PisG semisimple.gV “ tgv | v P V u G KU S ÓU U E G u P U Let be any simple -submodule of . Now, notice that for every , : u ¨ gV “ g ¨ pg´ ugq V “ gV is also a -submodule of , since for any , we have P1U looomooon gV W KU gV g´ W ´ g gV “ V gPG gV 1 KG S gV “ S U Moreover, is also simple, since if weregP aG non-trivial1 proper ř-submodule of then would also be a non-trivial proper submodule of . Now is non-zero and it is a ř G -submodule of , which is simple, hence S ÓU “ gV. Restricting to , we obtain that gPG ÿ G KU S ÓU

Remarkis a sum 18.2 of simple -submodules. Hence is semisimple. KU gV

The -submodules which appear in the proof of Theorem 18.1 are isomorphic to modules we g b V ÝÑ gV have seen before: more precisely the map g b v ÞÑ gv

g KU U E G U “ U U g b V gV is a -isomorphism, since implies that and hence the action of on (see Remark 17.1) and is prescribed in the same way. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Theorem 18.3 (Cliﬀord’s Theorem, strong form) 49 U E G S KG

G a ar Let be a normal subgroup andS letÓU “beS a‘ simple ¨ ¨ ¨ ‘ Sr -module. Then we may write 1 r P Zą S ;:::;Sr 1 KU a ; : : : ; ar 0 1 where and are pairwise non-isomorphic simpleG -modules, occurring with 1 G S Ó multiplicities respectively. Moreover, the following statementsU hold: a “ a “ ¨ ¨ ¨ “ a pS q “ ¨ ¨ ¨ “ pS q (i) the group permutesr the homogeneousK componentsK r of transitively; a G a S1– pS2 qÒ KG 1 H “ pS q (ii) H and dim dim G ; and 1 1 1 1 1 1 (iii) G as -modules, where Stab . Proof : S ÓU

ai ďThei ď factr that is semisimple andSi hence can be written as a direct sum as claimed follows from TheoremKU 18.1. Moreover, by the chapter on semisimplicity of ringsSi and modules, we know that for each ai ai G 1 theg P homogeneousG gpSi componentq “ pgSiq is characterisedgSi by PropositionKU 7.1: it isS theÓU unique largest -submodule which is isomorphic to a direct sum of copies of . ď j ď r gSi “ Sj ai aj gpSiNow,q Ď S ifj then , where is a simple -submodule of (see the proof ´ of the weak formK pSiq of “ Cliﬀord’sK pgS Theorem).iq Hencea therei ď aj exists an index 1 Sj “ g suchSi that aj ďandaj ai “ aj(alternatively to Proposition 7.1, the theorem of Krull-Schmidt can1 also be invoked here). Because dim dim , we have that . Similarly, since , we obtain . G G a ar Hence holds. Because S ÓU “ gpS ÓU q “ gpS q ‘ ¨ ¨ ¨ ‘ gpSr q ; 1 a G 1 gPG gpS q KG S gpSa q “ S 1 gPG ř we actually have that permutes the homogeneous components. Moreover,1 as 1 is a non- ř zero -submodule ofK , which is simple, we have that 1 , and so the action on the homogeneous components is transitive. This establishes both (i) and (ii). pSa qÒG “ KG b Sa “ g b Sa ÝÑ S For (iii), we deﬁne a -homomorphismH viaKH the map gPrG{H s 1 g b1m 1 ÞÑ gm 1 1 À 1 Φ : 1 1 1 gbm P gbSa KG K gpSa q S 1 G{H G 1 a a H 1 KG bKH S S |G H | K g b S1 that is, wherea . This is in fact a a-homomorphism. Furthermore, the -subspaces gpS q K 1 1 S g b m Ø m gm Ø m 1 of are in bijection with the cosets 1 , since permutes them transitively by (i), and the stabiliser 1 1 of one of them1 is . Thus both 1 and 1 are the direct sum of : KG-subspaces 1 and 1 respectively, each -isomorphic to 1 (via and ). Thus the restriction of Φ to each summand is an isomorphism, and so Φ itself must be bijective, hence a -isomorphism.

OneCorollary application 18.4 of Cliﬀord’s theory is for example the following Corollary: K G p G G p KG X ÒH X KH Assume is a splittingH ď G ﬁeld for of arbitrary characteristic. If is a prime number and is a -group, then every simple -module has the form , where is a 1-dimensional -module Prooffor : some subgroup . |G| |G| “ |G| G S We proceed byH induction“ G X “ onS . If 1 or is a prime number, then is abelian and any simple module is 1-dimensional by Corollary 12.3, so , and we are done. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

50 |G| “ pb b ą S KG

U “ tg P G | g ¨ x “ x @ x P Su : So assume with 1, and let be a simple -module and consider the subgroup G K S G : S “ G{U pT q K rG{Us T This is obviouslyU ‰ t u a normal|G{U subgroup| ă |G| of since it is the kernel of the -representation associatedH{U ď toG{U. G{U Hence Inf KforrH a{U simples Y -moduleT “. H{U pY q Now, if 1 , then , so by the induction hypothesis there exists a subgroup G G G{U G H and a 1-dimensional S “ -moduleG{U pT q “suchG{U that˝ H{UIndpY q “ H. But˝ H then{U pY q ;

H X “ H{U pYInfq Inf Ind Ind InfU “ t u G G so that settingG : Inf yields the result.A Thus we may assume 1 . IfA isZ abelian,pGq then all simple modules areG1-dimensional, so we are done.G Z AssumepG{ZpG nowqq that is not abelian. Then G{hasZpG aq normal abelian subgroupp thatx P isZ notpG central.qzZpGq Indeed,A to“ construct xZpGq; x thisy subgroup 2 , let denote the second centreZpGq of , that is, the preimage in of (this centre is 2 non-trivial as is a non-trivial -group). If , then : is a normal S ÓG“ Sa ‘ ¨ ¨ ¨ ‘ Sar abelian subgroup not contained in . Now,A applying Cliﬀord’sr Theorem yields: 1 a G r P Zą S ;:::;Sr 1 KA S “ pS q ÒH H “ a a GpS q V “ S KH 1 1 1 0 1 G 1 1 1 where , W WareÒH non-isomorphic simple -modules and S , where 1 1 1 H Stab . WeH argue‰ G that : 1 must be a simple V -module,“ X ÒH since if itH hadď H a properX non- trivial submoduleKH , then would be a proper non-trivial submodule1 of , which is simple: a 1 1 contradiction. If then by the induction hypothesis , where and is a S “ pSa qÒG “ pX ÒH qÒG “ X ÒG ; 1-dimensional -module. Thefore, by transitivityH of theH induction,H H we have 1 1 1 1 1 H “ G as required. 1 S ÓG“ S ÓH “ Sa ; Finally, the case cannot happen. ForA if it wereA to happen then 1 1 1 A A S is simpleG by the weak form of Cliﬀord’sS Theorem, henceA Ď ofZp dimensionGq 1 since is abelian. The elementsA of must therefore act via scalar multiplication on . Since such an action would commute with the Remarkaction 18.5 of , which is faithful on , we deduce that , which contradicts the construction of . p KG G This result is extremelyN ÒH useful, for example,H toď constructG the complex characterKH table of a -group. monomialIndeed, it says that we need look no furtherCG than induced linear characters. InM general,group a - modulep of the form M for some subgroup and some 1-dimensional -module is called . A group all of whose simple -modules are monomial is called an - . (By the above -groups are -groups.) Chapter 6. Projective Modules over the Group Algebra

We continue developing techniques to describe modules that are not semisimple and in particular indecomposable modules. The indecomposable projective modules are the indecomposable summands of the regular module. Since every module is a homomorphic image of a direct sum of copies of the regular module,Notation by knowing the structure of the projectives we gain some insightG into the structure of all modules.K

: throughout this chapter, unless otherwise specified, we let denote a finite group and be a field. (We will understand in the coming chapters,K why it is enough for our purposeKG to focus on ˝ fields, when considering projective module over the group algebra.) All modulesKG over groupKG algebras considered are assumed to be finite-dimensional over , hence finitely generated as -modules. References:When no confusion is to be made, we denote the regular module simply by instead of .

Local representation theory

[Alp86] J. L. Alperin. Representations and cohomology.. Vol. 11. I Cambridge Studies in Advanced Mathe- matics. Cambridge University Press, Cambridge, 1986. [Ben98] D. J. Benson. Methods of representation. Vol. theory. 30. Cambridge Vol. I Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1998. [CR90] C. W. Curtis and I. Reiner.Representations of groups . John Wiley & Sons, Inc., New York, 1990. [LP10] K. Lux andA H. course Pahlings. in ﬁnite group representation. Vol. theory 124. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010. [Web16] P. Webb. . Vol. 161. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016. 19 Radical, socle, head

Before focusing on projective modules, at this point we examine further theradical structure of socle-modules which are not semisimple, and try to establish connections with their semisimple submodules, semisimple quotients, and composition factors. This leads us to the deﬁnitions of the and the of a module.

51 Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Deﬁnition 19.1 52 M KG

radical M pMq “ V pMq Let be a -module. V P pMq KG M Ş (a) The of is its submodule rad : Max where Max denotes the set head M pMq “ M{ pMq of maximal -submodules of . socle M pMq M (b) The of is the quotient module hd : rad .

(c) The of , denotedK soc KGis the sum of all simple submodules of . M ‰ pMq ‰ M Notice that in our settingtop (i.e. isbottom a ﬁeld) -moduleshead are alwayssocle Noetherian, so provided 0 the above intersection is non-empty and hence rad . Also, informally (talks/spoken mathematics) Lemmaone also 19.2 uses the words and instead of and , respectively. M KG KG M

pMq Let be a -module. Then the following -submodules of are equal: JpKGqM (1) rad ; KG M (2) ;

Proof :(3) the smallest -submodule of with semisimple quotient.

“ V Ă M M{V V ;:::;Vr r P Zą M 1 0 “(3) (1)”: Recall that if is a maximal' submodule,M ÝÑ thenM{V ‘ ïs ¨ ¨ simple. ‘ M{Vr Moreover, if ( ) are maximal submodules of , thenm the mapÞÑ pm ` V ; : : : ; m ` Vrq 1 : KG p'q “ V X ¨ ¨ ¨1 X Vr M{pV X ¨ ¨ ¨ X Vrq – p'q M{ pMq 1 1 is a -homomorphism with kernel ker . Hence Im is semisimple,X Ď M sinceKG it is a submoduleM of{ aX semisimple module. Therefore rad is a semisimple quotient.KG It remains toX see;:::;X that itr isM ther smallestP Zą such quotient.X If is a -submodule with semisimple, then by the Correspondence Theorem, there M{X – X 1{X ‘ ¨ ¨ ¨ ‘ Xr{X 0 Xi{X @ ď i ď r : exists -submodules of ( ) containing such that ď i ď r 1Yi M M{X Xi{X and is simple 1 Yi Xi{X X “ Y X ::: X Yr X Ě pMq Ď For each 1 , let be theM kernel{JpKG ofqM the projectionKG{JpKG homomorphismq , so 1 that is maximal (as is simple) and . Thus rad , as required. JpKGq pM{JpKGqMq “ : “(1) (2)”: Observe that the quotient module is a -module as KG{JpKGq M{JpKGqM KG{JpKGq 0 KG Now, as is semisimple (by Proposition 6.6pM andq Proposition 6.7),KG is a semisim-M ple -module by definition of a semisimplepMq Ď JpKG ring,qM but then it is also semisimple as a - Ď module.Z Ď M Since weKG have already proved thatM{Zrad is the smallest JpKG-submoduleq ¨ M{Z “ of with semisimpleJpKGq quotient, we mustKG have that rad . JpKGqM Ď Z “(2) (1)”: If is any -submodule for“ which is semisimple,JpKGqM certainlyĎ pMq 0, because annihilates all simple -modules by definition, and it follows that . Thus, again as we already know that (1) (3), we obtain that rad . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Example 9 53 M KG pMq “ M pMq “ pMq “ M If is a semisimple -module, then soc by deﬁnition, rad 0 by the above Lemma, Lemmaand 19.3hence hd . M KG KG M

pMq Let be a -module. Prove that the following -submodules of are equal: KG M (1) soc ; tm P M | JpKGq ¨ m “ u (2) the largest semisimple -submodule of ;

Proof :(3) tm P M | J0pKG. q ¨ m “ u KG M JpKGq KG{JpKGq Exercise. [Hint: 0 is the largest -submodule of annihilated by , Remarkand hence 19.4 may (Socle, be seen radical as a and Loewy-module.] layers)

KG M n P Zě

2 We can iterate the notions of socle and radical: for each -module and each we deﬁne n n´ n n´ n´ inductivelypMq “ pMq pMq{ pMq “ pM{ pMqq ´ 1 ¯ 1 1 rad : rad rad pMq “ andpMq soc pMqsoc “ pMq : soc soc

1 1 where we understand that rad rad and soc soc . npMq “ JpKGqn ¨ M npMq “ tm P M | JpKGqn ¨ m “ u Exercise. Prove that: ¨ ¨ ¨ Ď pMq Ď pMq Ď pMq Ď M Ď pMq Ď pMq Ď pMq Ď ¨ ¨ ¨ (a) rad and soc 0 ; 3 2 radical series2 socle3 series M (b) rad rad rad and 0 soc soc soc M Loewy series M n´ pMq{ npMq n n´ The chains of submodulesradical layers in (b) areLoewy called layers respectively,M the andpMq{ 1 pMofq . The radicalsocle series layers of is alsoM known as the of . The quotients rad 1rad are called the , or of , and the quotients soc soc are Exercise 19.5 called the of . M N KG

n P Z npM‘Nq – npMq‘ npNq npM‘Nq – npMq‘ npNq Let and be -modules.ě Prove the following assertions. 1 M KG M (a) For every , rad rad rad and soc soc soc . M M r (b) The radical series of is the fastest descendingr n series of -submodules of with semisim-pMq “ n ple quotients,pUq “ andM the socler “ n series of is the fastest ascending series of with semisimple quotients. The two series terminate, and if and are the least integers for which rad 0 Deﬁnitionand 19.6soc then . KG M Loewy length M n JpKGqn ¨ M “ The common length of the radical series and socle series of a -module is called the of . (By the above, we may see it as the least integer such that 0.) Skript zur Vorlesung: Modular Representation Theory WS 2020/21

20 Projective modules 54

For the sake of clarity, we recall the general deﬁnition of a projective module through its most standard equivalentProposition-Deﬁnition characterisations. 20.1 (Projective module) R P R

pP; ´q R Let be an arbitrary ringR and let be an -module. Then the following are equivalent: R pP; ´q (a) The functor Hom is exact. In other words, the image of any s.e.s. of -modules under ψ P pM;Nq R Hom Ris again a s.e.s. ψ˚ R pP;Mq ÝÑ R pP;Nq (b) RIf Hom is a surjective morphism of -modules, then the morphism of abelian groups : Hom Hom is surjective.P In other words, for every pair of -morphisms α M N ψ ψ KG β P ÝÑ M α “ ψβ

π M ÝÑ P R π σ P where is surjective, there exists a -morphism : such that . R pP;Mq π ˝ σ “ P (c) If : is a surjective -homomorphism, then splits, i.e., there exists P R Hom such that Id . P P projective (d) The module is isomorphic to a direct summand of a free -module. PIM R If satisﬁes these equivalent conditions, then is called . Moreover, a projective inde- Example 10 composable module is called a of .

e R R – Re ‘ Rp ´ eq Re (a) Any free module is projective. e ‰ ; (b) If is an idempotent element of the ring , then 1 and is projective, tP u but not free if 0 1. i iPI Pi (c) It follows from condition (d) of Proposition-Deﬁnition B.5 that a direct sum of modules R is projective if and only if each is projective. R˝ R (d) If is semisimple, then on the one hand any projective indecomposable module is simple, and conversely, since is semisimple. It follows that any -module is projective.

21 Projective modules for the group algebra

We have seen that over a semisimple ring,projective all simple modules appear as direct summands of the regular module with multiplicity equal to their dimension. For non-semisimple rings this is not true any more, but replacing simple modules by the modules, we will obtain a similar characterisation. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

55 KG

To begin with we review a series of properties of projective -modules with respect to the operations onProposition groups and 21.1 modules we have introduced in Chapter 4, i.e. induction/restriction, tensor products, . . . K

P KG M KG Assume is an arbitrary commutative ring. Then the following assertions hold. K P bK M (a) If is a projective -module and is an arbitrary -module which is free of ﬁnite rank P KG H ď G P ÓG KH as a -module, then is projective. H H ď G P KH P ÒG KG (b) If is a projective -module and , then H is a projective -module.

(c) If and is a projective -module, then is a projective -module. Proof : P KG KG P1 n P ‘ P1 – pKGqn (a) Since is projective, by deﬁnition it is a direct summand of a free -module, so there exist a pKGqn b M – pP ‘ P1q b M – P b M ‘ P1 b M -module and a positiveK integer such thatK K . Therefore,K n pKGq bK M

and it suffices to prove that is free. So observe that Example7(a), Proposition 15.11(a) KG M K G M K M G G M G and thebK properties– p Òt ofuq the bK tensor– p productbK yieldÓt uqÒt u – Òt u K pMq G G K pMq K pMq 1 1 1 – pK 1 qÒt u– pK Òt uq – pKGq rk rk rk G 1 1 M Ót u M K n n¨ K pMq pKGq bK M – pKGq KG 1 since is just seenrk as -module,KH and, as such, is free of finite rank. It follows immediately that is a free -module, as required. KG ÓG – KH ‘ ¨ ¨ ¨ ‘ KH (b) We have already seen that as a -module,H G KH |G H| KG ÓH KH G H KG KH P | F G G KGwhere occursF withP multiplicityÓH | F ÓH : , so is a free -module. Hence the restriction from to of any free -module is a free -module. Now, by definition for some free G -module , soKH thatÒH – KG and the claim follows. (c) Exercise! [Hint: prove that .] KG KG

We now want to prove that the PIMs of are in bijection with the simple -modules, and hence that there are a ﬁnite number of them, up to isomorphism. We will then be able to deduce from this bijection that each of them occurs in the decomposition of the regular module with multiplicity equal toTheorem the dimension 21.2 of the corresponding simple module.

P KG P{ pPq KG M KG M{ pMq – P{ pPq KG (a) If is aP projective indecomposable -module,KG then rad' PisÝÑ a simpleM -module. M (b) If is a -module and rad M{ pradMq – Pfor{ ap projectivePq indecomposableM – P - module , then there exists a surjective -homomorphism : . In particular, if is also projective indecomposable, then rad rad if and only if . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

„ (c) There is a bijection KG {– ÐÑ KG {– projective indecomposable( simple ( -modulesP ÞÑ P{ -modulespPq

KGrad

Proof : and hence the number of pairwise non-isomorphic PIMs of is ﬁnite. P{ pPq KGpP{ pPqq

(a) By Lemma' P 19.2,KGpPradq is semisimple, hence it suﬃces to prove that it is indecomposable, or equivalently, by Proposition 5.4 that End rad is a local ring. 'p pPqq “ 'pJpKGqPq “ JpKGq'pPq Ď JpKGqP “ pPq : Now, if End , then by Lemma 19.2, we have ' KG rad rad ' P{ pPq ÝÑ P{ pPq

Therefore, by the universal property of the quotient,' induces a unique -homomorphism : rad rad such thatP the following diagramP commutes:

πP ö πP P{ pPq P{ pPq '

rad rad

KGpPq ÝÑ KGpP{ pPqq Then, the map ' ÞÑ ' Φ : End End rad K ψ P KGpP{ pPqq KG ' P ÝÑ P is clearlyψ ˝ π aP “-algebraπP ˝ ' homomorphism.ψ Moreover Φ is surjectiv. Indeed, if End rad , then by the deﬁnitionψ “ of' a projective module there exists a -homomorphism : such that P . But then satisﬁesKGpPq the diagram of the universal propertyKG ofpP theq quotient and by uniqueness . KGpP{ pPqq Finally, as is indecomposable End is local, hence any element of End is either nilpotent or invertible, and by surjectivity of Φ the same holds for End rad , which in turn must be local. P

(b) Consider the diagram πP M M{ pMq – P{ pPq πM ψ

πM πP rad P rad KG ' P ÝÑ M πP “ ψ ˝ πM ˝ ' where andM “ are'pP theq ` quotientpMq “ morphisms.'pPq ` JpKG AsqM is projective,'pPq “ byM definition, there exists a -homomorphismM : such that ' . It follows thatM | P rad , so that Mby– Nakayama’sP Lemma. Finally,M – P if is a PIM,M{ the surjectivepMq – P{ homomorphismpPq splits by definition of a projective module, and hence . But as both modules are indecomposable, we have . Conversely, if , then clearly rad rad S. KG S (c) The given map betweenKG the twoF sets is well-definedKG by (a) and (b), andψ it isF injectiveÝÑ S by (b). It remains to prove that it is surjective. So let Pbe aF simple -module.ψ|P P AsÝÑ Sis finitely generated, there exists a free -module and a surjective -homomorphism : . But then there is an indecomposable direct summand of such that : is non-zero, hence Skript zur Vorlesung: Modular Representation Theory WS 2020/21

57 S pPq Ď pψ|P q KG

surjective as Pis{ simple.pPq ÝÑ ClearlyS rad kerψ|P since itP is{ thep smallestPq -submoduleP{ pPq –withS semisimple quotient by Lemma 19.2. Then the universal property of the quotient yields a surjective homomorphism rad induced by . Finally, as rad is simple, rad Deﬁnitionby 21.3 Schur’s (Projective Lemma. cover of a simple module)

S KG PS KG S projective Ifcoveris aS simple -module, then we denote by the projective indecomposable -module Corollarycorresponding 21.4 to through the bijection of Theorem 21.2(c) and call this module the of . K G KG KG AssumeKG is a splitting ﬁeld for . Then, inK p theP{ decompositionpPqq of the regular module into a direct sum of indecomposable -submodules, each isomorphism type of projective indecompos- ns able -module occurs with multiplicityKGdim– radpPSq . In other words, with the notation of S Deﬁnition 21.3, à S simple KG nS “ K S where more preciely runs through the set of isomorphism classes of simple -modules and Proof : KG “ P ‘ ¨ ¨ ¨ ‘ P r P Z P ;:::P dim . r ą r Jp1KGq “ JpKGqKG “0JpKGqP ‘ ¨ ¨ ¨ ‘ JpKGqPr “ pP q ‘ ¨ ¨ ¨ ‘1 pPrq Let ( ) be such a decomposition. In particular, are PIMs. Then 1 1 rad rad KG{JpKGq – P { pP q ‘ ¨ ¨ ¨ ‘ Pr{ pPrq by Lemma 19.2. Therefore, 1 1 KG{JpKGq KG{JpKGq rad rad K where each summand is simple by Theorem 21.2(a). Now as is semisimple, by Theorem 8.2, any simple -module occurs in this decomposition with multiplicity equal to its -dimension. Thus the claim follows from the bijection of Theorem 21.2(c).

TheCorollary Theorem 21.5 also leads us to the following important dimensional restriction on projective modules. K G p ą P KG

Assume is a splitting ﬁeld for of characteristic|G|p K pPq :0. If is a projective -module, then ˇ ˇ dim |G|p p |G| p G

(Here is the -part of , i.e. the exact power of that divides the order of G.) Proof : Q P ppGq p G P ÓQ KQ KQ KQ Let Syl be aKQ Sylow -subgroup of . By Lemma|Q| “ 21.1 |G|p, is projective. Moreover, by Corollary 12.4 the trivial -moduleG is the uniquem simple -module, hence by Theorem 21.2(c) P Ó – pKQq m P Zą : has a unique PIM, namely itself,Q which has dimension . Hence 0 G for some K pPq “ K pP ÓQq “ m ¨ K KQ “ m ¨ |Q| “ m ¨ |G|p Therefore, dim dim dim and the claim follows. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

22 The Cartan matrix 58

KG S Now that weKG have classiﬁedM the projective -modules we turn to one of their important uses, which is to determine the multiplicity of a simple module as a composition factor of an arbitrary ﬁnitely M M M M M M generated -module (hence“ withĂ a compositionĂ Ă ¨ ¨ series). ¨ Ă n´ WeĂ recalln “ that if M 0 1 2 M1 M i n S 0 i{ i´ ď ď

1 is any composition series of S, theM number of quotients (1 ) isomorphic to is determined independently of the choice of composition series, by the Jordan–Hölder theorem. We call thisProposition number the22.1 multiplicity of in as a composition factor.

S KG PS T KG Let be a simple -module and let be its projective cover.

(a) If is a simple -module then K KGpSq S – T; K KGpPS;T q “ # S ﬂ T: dim End if dim Hom M KG 0 S if M

(b) If is an arbitrary -module,K thenKG thepP multiplicityS;Mq{ K of KGasp aSq composition: factor of is

dim Hom dim End Proof : M

DeﬁnitionExercise, 22.2 Sheet 4. [Hint: (b) can be proved by induction on the composition length of .]

S T KG

(a) If and are simplecST “-modules, thenS the integer PT

Cartan invariant: multiplicity of as a compositionpS;T q factor of C “ pc q is called the ST associated to the pair . KG Cartan matrix KG (b) The matrix : with rows and columns indexed by the isomorphism classes of simple -modules is called the of .

ItCorollary follows immediately 22.3 from Proposition 22.1 that the Cartan invariants can be computed as follows. S T KG

cST “ K KGpPS;PT q{ K KGpSq : If and are two simple -modules, then K G dim Hom dim End cST “ K KGpPS;PT q : In particular, if the ground ﬁeld is a splitting ﬁeld for , then dim Hom Skript zur Vorlesung: Modular Representation Theory WS 2020/21

59 KG decomposition matrix We will see later that there is an extremely eﬀective way of computing the Cartan matrix using another matrix associated to the simple -modules, called the . 23 Symmetry of the group algebra

KG pKGq˚ – KG KG We now want to obtain information about projective -modules using duality. Recall that we have already proved in Lemma 15.9 that as -modules. This allows us toProposition deduce that 23.1 projectivity is preserved by taking duals. P KG P P˚

Proof : P˚˚ – P KG P If is an -module, then˚ is projective if and only if n ˚ is. ˚ n n pKGq n P Zě P ppKGq q – ppKGq q – pKGq As as -modules it suﬃces to prove one implication. Now, if is a direct summand 1 of ( ), then is a direct summand of , and so is also projective.

Next we want to investigate the relationship between head and socle of projective modules. For this purpose, we recall the following˚ properties of submodules, quotients and duality,˚ seen in˚ the Exercise˚ classes:W ď V KG ñ V KG M M – pV {W q V {M – W

-submodule has a -submodule such that and

Corollary 23.2 KG P

˚ ˚ ˚ Every projective indecomposable -modulepPq – pPhas{ a simplepP qq socle.: More precisely,

soc rad Proof : P˚{ pP˚q pP˚q KG P˚ ˚˚ KG As on the oneP hand– P rad is simple by TheorempPq 21.2 and on the other hand rad is the smallest -submodule of with semisimple quotient by Lemma 19.2, its dual is the largest semisimple -submodule of P , hence isomorphicKG to soc , which has toP˚ be simple, as required. SAlternatively, weKG could argue that as the socle isKG by deﬁnition the sumS ÝÑ of allP simple submodules, it ˚ ˚ suﬃcesKG to prove that hasP aÝÑ uniqueS simple -submodule. Because is projective by Proposition 23.1, if is any simple -module, then by duality the -homomorphisms are in bijection with pS;Pq “ pP˚;S˚q the -homomorphisms K andKG it follows that K KG S˚ dim Hom dim Hom

˚ ˚ ˚ Moroever is also simple.˚ Thus˚ it followsK fromKG PropositionpS q P 22.1(a) that S ; K KGpP ;S q “ # : dim End if is the projective cover of dim Hom ˚ 0 K KGelsepS q “ K KGpSq

Therefore, the claim follows from the fact that dim End dim End (again by duality). Skript zur Vorlesung: Modular Representation Theory WS 2020/21

˚ ˚ ˚ 60 K KGpS;Pq “ K KGpP ;S q K KGpS q “ K KGpSq Notice that in this proof to see that dim Hom dim Hom and dim End dim End it is also possible to argue using Lemma 13.4 and Lemma 14.2. symmetric algebra In fact, we can obtain a more precise statement and prove that the head and the socle of a PIM are isomorphic.Remark 23.3 For this purpose, we need the fact that the group algebra is a .

p ; q G ˆ G ÝÑ K; pg; hq “ δg;h´ The map K 1 : p ; q KG ˆ KG Ñ K: extended byK -bilinearity to pab; cq “ pa; bcq @ a; b; c P KG : deﬁnes a -bilinearK form, which is symmetric, non-degenerate and associative. (Associative means thatK symmetric). algebra More generally, a -algebra endowed with such a symmetric, non-degenerate and associative Theorem 23.4 -bilinear form is called a . P KG P{ pPq – pPq

Proof : S “ P{ pPq P KG S ﬂ pPq If is a projective indecomposablen -module, then rad soc . KG “ R ‘ Q Q “ P n P Zą KG Put : radP , whichP - R we know is simple as is a PIM of , and assume soc . Write 0 n , where ( ) is thep directQq – sumpP ofq all the indecomposable direct summands of isomorphicQ to and KG. Then S KG KG S I soc soc ‰ I Ă R I KG andψpIqdoes Ď I not containψ anyP KG-submodulepKGq I isomorphic toKG . Next, considerJ “ t theψ P sum ofKG allpKGq |-submodulespψq Ă Iu of isomorphicJ to and denote itKG bypKG, soq thatπ clearlyKG 0Ñ Q and is a left idealQ of . However,R as for every End , is an ideal of . Now set : End Im , so that is clearly an ideal of End ' P J.ñ Let pKG: q Ď p'beq the projection onto with kernel . Then: ' KG I KG rad ker '|R “ S - pRq as the image of is semisimple, because it is a -submodule of , which is itself a -submodule. ' ˝ π “ ' π ˝ ' “ Thus, 0, as hd , and it follows that 'pKGq Ď I und 0 ' “ ' ˝ π ´ π ˝ ' @ ' P J: p˚q

(as ‰ ')P andJ hence S “ pPq Ď pKGq α P KGpKGq ' ˝ α P J 'α “ 'απ ´ π'α ˚ Let now 0 (exists since hd hd ), and let End , so and a “ αp q b “ 'p q c “ πp q by ( ). ab “ αp q'p q “ 'pαp qq “ 'pαpπp qqq ´ πp'pαp qqq “ cab ´ abc Set then : 1 , : 1 , : 1 , so

1 1 1 1 1 pa; bq “ pab; q “ pcab; q ´ pabc; q “ pc; abq ´ pab; cq “ : and it follows from Remark 23.3 that α P pKGq a P KG p ; q KG 1 1 1 0 b “ ' “ This is true for every End , and hence every . Finally, as is non-degenerate, we have 0 and hence 0. Contradiction! Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Corollary 23.5 61 S KG

P KG S P{ pPq Let be a simple -module. S pPq

(a) If is any projective -module,G then the multiplicity of˚ G in rad ˚ equals the multiplicity of in soc . In particularK pP q “ K pPGq “ K pP q “ K pP qG :

˚ pPSq – PdimS˚ dim dim dim

Proof :(b) We have . KG

G ˚ G ˚ (a) By Theorem S23.4“,K the ﬁrst claim holds for the PIMsK pP q of “ , henceK pPGq this “ is alsoK pP trueq “ for anyK ﬁnitepP qG direct: ˚ sum of PIMs, because taking socles and radicalsp commutePSq with the direct sum by Exercise 19.5(a). ˚ Nextp pP takingSqq yields the equalities dim dim dim dim (b) We have seen in the proof of Corollary 23.2˚ that is the˚ projective˚ cover of the simple module p pPSqq – pPS{ pPSqq – S : soc . Moreover, by Theorem 23.4 ˚ pPSq – PS˚ soc rad Hence .

Finally, we see that the symmetry of the group algebra also leads us to the symmetry of the Cartan matrix.Theorem 23.6 S T KG

If and are simple -modules,cST ¨ K then KGpSq “ cTS ¨ K KGpT q :

K dim EndG dim End KG Proof : In particular, if is a splitting ﬁeld for , then the Cartan matrix of is symmetric. cST “ K KGpPS;PT q{ K KGpSq By Corollary 22.3,

cTS “ dimK HomKGpPT ;PSq{ dimK EndKGpT q ;

and K KGpPS;PT q “ K KGpPT ;PSq dim Hom dim End G ˚ G so it is enough to prove that KGdimpPSHom;PT q “ K pPS;PdimT q Hom– ppPSq bK P. T q Now, by Lemma 14.2 and Lemma 13.4 we have

G ˚ G HomKGpPT ;PSq “ HomK pPT ;PSq – ppPT q bK PSq : ˚ and pPSq bK PT Hom ˚ HomG ˚ ˚ G K pppPSq bK PT q q “ K pppPSq bK PT q q q : Moreover, as is projective by Proposition 21.1(a), it follows from Corollary 23.5(a), that ˚ ˚ ˚ ˚ ppPSq bK PT q – PS bK pPT q – pPT q bK PS K KGpPS;PT q “ K KGpPT ;PSq dim dim But K G , thus we haveKGpS provedq – K that– dim KGHompT q dim Hom . cST “ cTS Finally, if is a splitting ﬁeld for , then by deﬁnition End End , so that the dimension of both endomorphism algebras is one and we have and we conclude that the Cartan matrix is symmetric. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

24 Representations of cyclic groups in positive characteristic 62

n G “ Zn “ xg | g “ y n P Zě K 1 We now describe the representations of a cyclic group : 1 of order over a aNotation: ﬁeld of positivep “ characteristic.pK q ą n “ p m a P Zě pp; mq “ K G K m 0 Set : charζm 0 and write with and gcd 1. More- over, we assume that is a splitting ﬁeld for , so it follows that contains a primitive -th root of Theoremunity, which 24.1 we denote by . This enables us to use the theory of Jordan normal forms (Linear Algebra).

n KZn n There are exactly isomorphism classes of indecomposable -modules. These correspond to the ζi matrix representations m i » ζm fi a Ri;r G Ñ rpK q; g ÞÑ 1 p ď i ď m; ď r ď p q: — . ffi — .. i ffi — ζm ffi : GL — . ffi 1 1 – .. 1 fl

a Proof : Ri;r ď i ď m; ď r ď p G “ Zn n r Ri;rpgq “ Ir pe ; : : : ; erq K K Zn First notice thatxe ; : : :( ;1 ej yK ď1 j ď r ) deﬁnes a matrixR representationi;r of sinceď a 1 i ď m; ď r .ď Furthermore,p if is the standard -basis of , then the only -invariant 1 subspaces are the Ri;r with 1 . As they form a chain, is indecomposable for all 1K 1 , because it cannot be writteng as the direct sum of two non-trivial subrepresentations. a It is also clear that the Ri;rare pairwiseď i ď m; non-equivalent,ď r ď p as they are uniquely determinedKZ throughn - dimension and eigenvalues at evaluation in . M KG K pMq “ r P Zą It remains to prove that theK (1 1 ) accountM for all the indecomposable -modules.R 0 We knowR frompgq the theory of Jordan normal form, that if is a -module with dim : M , then choosing a suitable -basis, we may assume that corresponds to a matrix representation such that is a block diagonal matrix where each block is a Jordan block. Assuming now that is Rpgqn “ Rpgnq “ Rp q “ I ; indecomposable, then there can be only one Jordan block. Moreover,G r as n K i 1 ζm ď i ď m ζm pK q “ p the eigenvalues (i.e. the diagonal entries of the Jordan blocks) can only be -th roots of unity in , and Rpgq “ su “ us hence they are powers (1 ) of since char . Furthermore, i i s “ pζm; : : : ; ζmq u

s s pu ´ I qp “ up ´ I “ @ ps ě r with diag and is the Jordanr block withr diagonal entries equal to 1, where it holds that u p Rpgq ď r ď pa 0 ,

so is the -part of and it follows that 1 . Corollary 24.2

¨ KZn R ď i ď m Up to isomorphism: i;

¨ KZ R a ď i ď m the simple -modulesn correspond precisely to the matrix representationsi;p 1 (1 );

the PIMs of correspond precisely to the matrix representations (1 ). Skript zur Vorlesung: Modular Representation Theory WS 2020/21

63 Proof :

¨ Ri; ď i ď m We can bound the number of modules in both families of modules as follows. KZn m 1 Firstly,KZ then matrix representations (1 ) all have degree 1, hence they must be irreducible a ¨ and correspond to simple -modules.P It followsKZ thatn therep are| atK leastpPq pairwise non-isomorphic simpleP -modules. Ri;pa ď i ď m Secondly,m by Corollary 21.5, if is a PIM of , then dim . Hence, up to equivalence, corresponds to one of the matrix representations with 1 . It follows that there are KZn at most pairwise non-isomorphic PIMs. KZn However, we know from Theorem 21.2(c) that there is a bijection between the simple -modules and the PIMs of . The claim follows. Chapter 7. Indecomposable Modules

wild representation type After simple and projective modules, the goal of this chapter is to understand indecomposable modules in general. Apart for exceptions, the group algebra is of , which, roughly speaking, means that it is not possible to classify the indecomposable modules over such algebras. However, representation theorists have developed tools which enable us to organise indecomposable modules in packages parametrised byrelative parameters projectivity that are useful enough to understand essential propertiesvertices of thesesources modules. In this respect, first we will generalise the idea of projective modules seen in Chapter 6 by defining what is called . This will lead us to introduce the concepts of and of indecomposable modules, which are two typical examples of parameters bringing us usefulNotation information about indecomposable modules in general. G K free of finite rank as K -modules: throughout this chapter, unless otherwise specified, we let denote a finite group and be a commutative ring. All modules over group algebras considered are assumed to be References: .

Local representation theory

[Alp86] J. L. Alperin. Representations and cohomology.. Vol. 11. I Cambridge Studies in Advanced Mathe- matics. Cambridge University Press, Cambridge, 1986. [Ben98] D. J. Benson. Methods of representation. Vol. theory. 30. Cambridge Vol. I Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1998. [CR90] C. W. Curtis and I. Reiner.Representations of groups . John Wiley & Sons, Inc., New York, 1990. [LP10] K. Lux and H.G Pahlings.-algebras and modular representation. Vol. theory 124. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010. [Thé95] J. Thévenaz.A course in ﬁnite group representation theory . Oxford Mathematical Mono- graphs. The Clarendon Press, Oxford University Press, New York, 1995. [Web16] P. Webb. . Vol. 161. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016. 25 Relative projectivity

The relationship between the representations of a group and those of its subgroups are one of the most important tools in representation theory of ﬁnite groups. In the context of modular representation theory this relationship shows itself in a reﬁnement of the notion of projectivity, namely relative projectivity. 64 Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Deﬁnition 25.1 65 H ď G

KG M H-free M – V ÒG KH V Let . H KG M relatively H-projective H-projective (a)A -module is called if for some G -module . H M | V ÒH KH V (b)A -module is called , or simply , if it is isomorphic Remark 25.2to a direct summand of an -free module, i.e. if for some -module . H

It is easy to see that -freeness is a generalisation of freeness and relative projectivity is a ¨ t u KG – K ÒG pKGqn – generalisation of projectivity. Indeed, t u n G pK qÒt u freeness is the same as 1 -freeness: as 1 by Example7, clearly ¨ t u KG ô 1 ; KG ô t u KG ô tprojectivityu is the same as 1 -projectivity: a -module is projective it is a direct summand of a free -module it is a direct summand of a 1 -free -module it is relatively 1 -projective.

To begin with, we would like to characterise relative projectivity in a similar way we characterised projectivity in Proposition-Deﬁnition B.5. To reach this aim, we ﬁrst take a closer look at the adjunction Notationbetween induction 25.3 and restriction, we have seen in Theorem 15.10. H ď G

' U ÝÑ U KH ' ÒG KG Let . H 1 2 (1) Let : G be a -homomorphism.G Then we denoteG by the induced - homomorhpism' ÒH “ KG b ' U ÒH “ KG bKH U ÝÑ U ÒH “ KG bKH U x b u ÞÑ x b 'puq: 1 1 2 2 : Id : KG

U KH V KG K obtained by tensoring with the identity morphism on . G – G (2) Let be a -module“ and U;V be a KG-module.pU ÒH ;V q TheÝÑ -isomorphismKH pU;V ÓH q

Φ: Φ : Hom G – Hom G “ U;V KH pU;V ÓH q ÝÑ KGpU ÒH ;V q and G H G Ψ: Ψ : Hom Hom H a pair of adjoint functors G G we obtained in Theorem 15.10 hides the factH thatleft the adjoint induction andH restriction functors Ind and Res form what is called in category theory. More precisely, this isomorphism translatesnatural the factU thatVInd is to Res . Explainedγ P KH inpU more;U q details, there may of course be many such isomorphisms, but there is a choice which is called in and . Spelled out, this means that whenever a morphism 1 2 Hom is given, the diagram Skript zur Vorlesung: Modular Representation Theory WS 2020/21

66 G U ;V G KGpU ÒH ;V q – KH pU ;V ÓH q 1 ˚ Φ G γ˚ pγÒH q 1 1 Hom Hom G – G KGpU ÒH ;V q KH pU ;V ÓH q U ;V 2 2 Hom Φ 2 Hom α P KGpV ;V q

1 2 commutes and whenever Hom G isU;V given, the diagramG KH pU;V ÓH q – KGpU ÒH ;V q Ψ 1 α α ˚ 1 ˚ 1 Hom Hom G – G KH pU;V ÓH q KGpU ÒH ;V q U;V 2 2 Hom Ψ 2 Hom ˚

commutes.unit (For the uppercounit and lower notation, see again Proposition D.3.) H Furthermore, such naturalKH isomorphisms depend on certain distinguished homomorphisms

called the and the G ofG the adjunction. In order to understand relative -projectivity, we need to considerµ U ÝÑ the U ÒH-homomorphismÓH “ g b U “ b U ‘ g b U gPrG{Hs gPrG{Hs;g‰ à à u ÞÑ b u : 1 1

1 U b U KG G G G (i.e. the natural inclusion ofε VintoÓH Ò theH “ summandg b1 pV Ó)H andq ÝÑ theV -homomorphism gPrG{Hs à : g b v ÞÑ gv

G unit counit H left G adjoint H u P U ε ˝ µpuq “ εp b uq “ u whichε ˝ µ “ are,U respectively, the and the of the adjunction saying that Ind is ¨ µ to ResKH . Further, weε note that for any , we have 1 , so Id and thus we deduce that: ¨ µ is a -section for ; ¨ ε is injective; and K is surjective. G G This allows us to“ constructU;V mutuallyKGpU ÒH inverse;V q ÝÑ naturalKH p-isomorphismsU;V ÓH q; ψ ÞÑ ψ ˝ µ ; G G G Φ “ ΦU;V : HomKH pU;V ÓH q ÝÑ Hom KGpU ÒH ;V q; β ÞÑ ε ˝ β ÒH :

Ψ Ψ : Hom Hom as required. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Proposition 25.4 (Characterisation of relative projectivity) 67 H ď G U KG

KG U H Let . Let be a -module. Then the following are equivalent. ψ U ÝÑ W KG ' V W U (a) The KG-module is relatively -projective. KH D α G G G G ψ (b) αIfH :U ÓH ÝÑ V isÓH a -homomorphism,' ˝ αH “ ψ : U ÓH is a sur- ö KG αG U ÝÑ V ' ˝ αG “ ψ jective -homomorphism and there exists a -homomorphism V ' W : such that on , then there exists a -homomorphism : such that so ' V U KG that theG diagramG on the right commutes. ' V ÓH ÝÑ U ÓH KH ' KG (c) Whenever : is a surjective -homomorphism such that the restriction KG : splits as -homomorphism, then splits as a -homomorhpism. G G (d) The surjective -homomorphismU ÓH ÒH “ KG bKH U ÝÑ U x b u ÞÑ xu

KG U U ÓG ÒG is split. H H KG N U | K ÒG b N (e) The -module is a direct summand of H . K

(f) There exists a -module such that . Proof : G ñ U “ T ÒH KG G ψ T ÒH ÝÑ W ' V W G G G (a) (b): First we consider the case where isKH an induced module.αH SupposeT ÒH ÓH ÝÑ thatV weÓH have - ψhomomorphisms“ ' ˝ αH : and : as shown in the diagram shown on the left below. Suppose, moreover, that there exists a -homomorhpism : such that G G G , that is, the diagramT ÒH on the right below commutes: T ÒH ÓH D α ψ H ψ ö V W G G ' V ÓH ' W ÓH

G G G G G µ T ÝÑ T ÒH ÓH ε V ÓHÒH ÝÑ V H G H µ KH ε KG Let : and : µbe the unit and the counit of the adjunctionKH of Res and Ind KHas deﬁned in Notation 25.3, so is an injective -homomorphism and is a surjective - homomorphism. Then, precomposing with , we obtain that the following triangle of -modules T and -homomorphisms commutes: αH ˝µ ψ˝µ ö G G V ÓH ' W ÓH

' V ÝÑ W KG

By the naturality of Φ and Ψ from Notation 25.3, since : is a -homomorphism, we have the following commutative diagram: Skript zur Vorlesung: Modular Representation Theory WS 2020/21

68 G T ;V G KH pT;V ÓH q – KGpT ÒH ;V q Ψ '˚ '˚ Hom Hom G – G KH pT;W ÓH q KGpT ÒH ;W q T ;W

Hom Ψ Hom

p' ˝ pαH ˝ µqq “ ' ˝ p pαH ˝ µqq : In other words, pψ ˝ µq “ p pψqq “ ψ Ψ Ψ

By the commutativity of the previousψ “ triangle,' ˝ ε ˝ pptheα left˝ µqÒ handG q side of this equation is equal to Ψ Ψ Φ since Ψ and Φ are inverseH to oneH another. Thus KG

T ÒG and so the triangle of -homomorphisms H G ε˝ppαH ˝µqÒ q H ψ ö

V ' W

G U “ T ÒH G ι G π U T ÒH U ÝÑ T ÒH ÝÑ U G G commutes, proving the implication forKH an inducedαH module.U ÓH ÝÑ V ÓH Now let be any direct summand of . Let denote the canonical inclusion and U ÓG projection. Suppose that there is a -homomorphism H : such that the diagram D αH ψ ö G G V ÓH ' W ÓH

G ' ˝ αH “ ψ U ÓH

G T ÒH commutes, i.e. G on . Then we considerT ÒG ÓG the following diagrams: T ÒH H H αG ψ˝π αH ˝π ö ψ˝π ψ˝π ö V ÓG W G G H ' V ' W V ÓH ' W ÓH

KH αH G KG αG T ÒH ÝÑ V ' ˝ αG “ ψ ˝ π TheKG middle diagram of -homomorphisms commutes by deﬁnition of , and hence by the ﬁrst part there' ˝ αG is˝ aι “ ψ-homomorphism˝ π ˝ ι “ ψ : such that , so the third diagram of -homomorhpisms also commutes. U Now , so the triangleα ˝ι G ψ ö

V ' W

ñ ' V U KG KH αcommutes,H KH as required. ' KH (b) (c): Let : be a surjective -homomorphism which is split as a -homomorphism, and let be a -section for . Thus, we have the following commutative diagram of -modules: Skript zur Vorlesung: Modular Representation Theory WS 2020/21

G 69 U ÓH αH ö U G G V ÓH ' U ÓIdH

KG αG U ÝÑ V ' ˝ αG “ U αG KG ' G G G G ñ Then assumingµ U ÝÑ (b)U isÓ true,H ÒH thereKH exists a -homomorphismε U ÓH ÒH ÝÑ: U such that Id . In particular, is a ε-section for KG. (c)ñ(d): Since : is a -section for : (see Notation 25.3), applying condition (c) yields that splits as a -homomorhpism,G and henceG (d)G holds. G G ñ K ÒH bK N – pK bK N ÓH q ÒH – N ÓHÒH (d) (e): isN immediate.“ U G G G (e)ñ (f): Recall that by Proposition 15.11 weK ÒH havebK N – N ÓHÒH . Thus, setting : yields the claim. (f) (a): straightforward from the fact that seen above. Exercise 25.5 H ď J ď G U KG V KJ

U H U J Let . Let be a -module and let be a -module. Prove the following statements. U V ÒG V H U H (a) If is -projective then is J-projective. g P G U H gU gH (b) If is a direct summand of and is -projective, then is -projective. U H W KG (c) For any , is -projective if and only if is -projective. U bK W H (d) Using part (f) of Proposition 25.4, prove that if is -projective and is any -module, then is -projective. KG

RemarkProjectivity 25.6 relative (Projectivity to a subgroup relative can to beKG generalised-modules) as follows to projectivity relative to a -module:

V KG KG M projective relative to the module V relatively V -projective V -projective KG N M (a) Let be a -module. A -moduleV bK N is termedM | V bK N or pV q , or simply V if thereKG exists a -module such that is isomorphic to a direct summand of , i.e. . H ď G We let Proj denote the class of allG -projective -modules. KG V “ K ÒH (b) Proposition 25.4(f) shows that projectivity relative to a subgroup is in fact projectivity relative to the -module : .

Note that the concept of projectivity relative to a subgroup is proper to the group algebra, but the concept of projectivity relative to a module is not and makes sense in general over algebras/rings.

The following exercise provides us with some elementary properties of projectivity relative to a module, which also hold for projectivity relative to a subgroup, by part (b) of the remark. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Exercise 25.7 70 K p ą A;B;C;U;V KG

Assume is a ﬁeld of characteristcV 0 andKG let V be -modules. Prove that:

(a) AnyU P directp summandV q of apU-projectiveq Ď pV q -module is -projective;

(b) If p - ProjK pV q, then ProjKG Proj V;

˚ (c) If pVdimq “ thenpV q any -module is -projective;

(d) ProjpU ‘ VProjq “ p;Uq ‘ pV q

(e) ProjpUq X pProjV q “ pProjU bK V;q n m (f) Projp j“ ProjV q “ pProjV q “ p j;“ V q @ m; n P Zą

À Â 0 (g) CProj– A ‘ B1 V Proj Proj 1 A B ; V

˚ (h) pV q “ is pV-projectivebK V q if and only if both and are -projective;

(i) Proj Proj .

Hint: you may want to use Lemma 13.8 and Exercise 4(c) on Sheet 3. Proceed in the given order.

After this small parenthesisKG on projectivity relative to modules, we come back to projectivity relative to subgroups. We investigate further what information this concept brings to the understanding of indecomposable -modules in general.KG G Next we see that any indecomposable -module can be seen as a relatively projective module with respectTheorem to 25.8 some subgroup of . H ď G

|G H| K KG H Let . K p ą H p G (a) If : is invertible in , then every -module is -projective. KG H (b) In particular, if is a field of characteristic 0 and contains a Sylow -subgroup of , Proof : then everyV -moduleKG is -projective. V H V ' U V KG KH (a) Let be a -module. To prove' that is KG-projective, we prove that satisfies The- orem 25.4σ (c).V ÝÑ SoU let KH: be a surjective' -homomorphism which splits as a - homomorphism. We need toσ proveV thatÝÑsplitsU as a -homomorphism. So let : be a -linear section for and set ´ v ÞÑ |G H| gPrG{Hs g σpgvq r : ˆ 1 1 1 |G H| |G H| P K : ř σ g P G v P V . We may divide by : since : and clearly r is well-defined. Now, if and σpg1vq “ g´ σpgg1vq “ g1 pgg1q´ σpgg1vq “ g1σpvq , then |G H| |G H| gPrG{Hs gPrG{Hs 1 ÿ 1 1 ÿ 1 r r : : Skript zur Vorlesung: Modular Representation Theory WS 2020/21

' KG 'σpvq “ ' g´ σpgvq “ g´ 'σpgvq “ g´ gv “ v and |G H| |G H| |G H| gPG -lin. gPG gPG 1 ÿ ` 1 ˘ 1 ÿ 1 1 ÿ 1 r : 'σ: “ V σ KG : ' ˆ P P ppGq H Ě P p - |G H| |G H| P K where the last-but-one equality holds because Id . Thus r is a -linear section for . (b) follows immediatelyH “t fromu (a). Indeed, if Syl and , then : , so : .

Considering the case 1 shows that the previous Theorem is in some sense a generalisation of Maschke’sRemark 25.9 Theorem (Theorem 11.1). K p ą H “ t u H p G p G p - |G| Assume that is aKG field of characteristict u 0 and 1 is the trivial subgroup. If contains a Sylow -subgroup of then thep Sylow- |G| -subgroupsKG of are trivial, so . The theorem then says that all KG-modules are 1 -projective and hence projective. We know this already, however! If then is semisimple by Maschke’s Theorem (Theo- Corollaryrem 11.1 25.10), and so all -modules are projective by Example 10(d). H ď G |G H| K KG U G U ÓH Let and suppose that : is invertible in . Then a -module is projective if and only if is projective. K p ě H p G Again, this holds in particular if is a field of characteristic 0 and contains a Sylow -subgroup ofProof. : G U ÓH The necessary condition is given by Proposition 21.1(b). To prove the sufficient condition, suppose U ÓG | pKHqn n P Z : that is projective. Then, on theH one hand, ą U H 0 for some U | U ÓGÒG : On the other hand, is -projective by Theorem 25.8H ,H and it follows from Proposition 25.4(e) that

G G n G n U | U ÓHÒH | pKHq ÒH – pKGq ; HenceU

so is projective. 26 Vertices and sources

K pK q “ p ą For the remainder of this chapter, we assume that is a ﬁeld with char : 0.

As said before, we now want to explain some techniques that are available to understand indecomposable modules better. Vertices and sources are two parameters making this possible. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Theorem 26.1 72 U KG

Q G Let be an indecomposable -module. U Q (a) There is a unique conjugacy class of subgroups of which are minimal subject to the Q G U Q property that is -projective. KQ T NGpQq G (b) ULet be a minimal subgroupT ÒQ of such thatKQ is -projective.T Then, there exists an inde- G composableU ÓQ -module which is unique, up to conjugacy by elements of , such that is a direct summand of . Such a -module is necessarily a direct summand of . Proof : U H K H K G

G G (a) Suppose that is both - and -projectiveU | U forÓH subgroupsÒH and of . Then, by Proposi- tion 25.4(e), G G G G K U U ÓH ÒH ÓK ÒK

and the latter module is -projective. Hence is also a direct summand of . By the G G G G G G G G Mackey formula and transitivityU ÓH ÒH ÓK Ò ofK “ inductionU ÓH andÒH Ó restriction,K ÒK it follows that ´ ` ˘ g ¯ G gH K G “ U ÓH ÓK X gH ÒK X gH ÒK ´ gPrK zG{Hs ¯ à g` ` G ˘ G ˘ “ U ÓK X gH ÒK X gH : gPrK zG{Hs à ` ˘ g P G U K X gH U K X gH K H g g Therefore, by the Krull-SchmidtU Theorem, there exists such thatK X His“ aK direct summandK Ď H of g´ a moduleH Ď inducedK fromH K , andG hence is -projective. Now, if both and are 1 G G minimalQ such that is projective relative to theseU subgroups, then U | U ÓQ.Ò Thus,Q and , hence and are -conjugate.G G T ÒQ T U ÓQ 1 1 G 1 G G (b) LetT be a minimal subgroup relativeKQ to which is projective.U | T ÒQ Then T | T ÒQÓQby Proposi- tion 25.4(e) so it is a direct summand of for some indecomposable direct summand of . 1 G G g 1 gQ Q If is another indecomposableT ÒQÓQ-module“ suchp T thatÓQX gQqÒQX gQ ,; then . Mackey’s formula says that gPrQzG{Qs à g P G

g 1 gQ Q hence, again by the Krull-Schmidt Theorem,T | p T thereÓQX g existsQqÒQX gQ such that T Q X gQ U Q g U Q “ Q X Q g P NGpQq T g 1 1 and therefore T is g-projective,P G andT henceT so is . Since is a minimal subgroup relative g 1 toT “ whichT isT projective, and hence NG.pQ Itq follows that is actually a direct g 1 G 1 g´ summandT “ ofT , for this . Since and areU indecomposable,ÓQ however,T “ thisT means that 1 , so g´is uniqueG up to conjugacyg´ by elements of . 1 p UqÓQ U – U KG T Now isG1 an idecomposable direct1 summand of by deﬁnition, so is a direct U ÓQ summand of . However, as -modules, so this means that is also a direct summand of .

This characterisation leads us to the following deﬁnition: Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Deﬁnition 26.2 73 U KG

vertex U Q G U Q Let be an indecomposable -module. U pUq (a)A of is a minimal subgroup of such that is relatively -projective. Q U KQ-source source U KQ T The setG of all vertices of is denoted by vtx . U | T ÒQ (b) Given a vertex of , a , or simply a of is a -module such that . Remark 26.3

KG U t u (a) Conceptually, the closer the vertices of a module are to the trivial subgroup, the closer this module is to being projective: a -module with trivial vertex is 1 -projective and hence Q KG U projective. U up to G-conjugacy (b) A vertex of an indecomposable -modulenever is not uniquelythe deﬁned, in general. However, the vertices of are unique , so inQ particularis vertex are of allU isomorphic. For thisthe verticesreason, in of general,U one (i.e. you!) should talk about vertex of a module (of course, unless a vertex has been ﬁxed). We either say that a , or talk about . (Unfortunately many textbooks/articles by non-experts are very sloppy with Q U U N pQq this terminology, inducing errors.) G

ToProposition begin with, 26.4 we have the following important restriction on the structure of the vertices. KG p G

ProofThe : vertices of an indecomposable -moduleKG are -subgroups of . p G p p By Theorem 25.8, we know that every -module is projective relative to a Sylow -subgroup of . Therefore, by minimality, vertices are contained in Sylow -subgroups, and hence are themselves Warning-groups.

: vertices and sources are very useful theoretical tools in general, but extremely diﬃcult to compute concretely.

WeExample show 11here how to compute the vertices and sources of the trivial module.

KG K p G pK q “ ppGq

The vertices of the trivial -module are the Sylow -subgroups of , i.e. vtx Syl , and all sources are trivial. Claim: P p H ď P K ÒP KP To establish this fact, we need the following indecomposabilityH property: Indeed: P p KP If is a P -group and , then is an indecomposable -module. K ÒH First recall that as is a -group, the only simple -module is the trivial module. Hence the socle of is a direct sum of trivial submodules. This together with Frobenius reciprocity Skript zur Vorlesung: Modular Representation Theory WS 2020/21

P P P yieldsK pK ÒH q “ K KP pK;K ÒH q “ K KH pK ÓH ;K q “ K KH pK;K q “ : K P K P U V KP dimÒH soc dim Hom dim Hom ÒdimH “ Hom‘ 1 U;V ‰

If were decomposable,P then so would be its socle: clearly, if for some - modules K pK0Ò,H thenq “ K p pUq ‘ pV qq “ K pUq ` K pV q ě ` “ : K P dim soc dim ÒHsoc soc dim soc dim soc 1 1 2

A contradiction!Q P p ThereforeK q P P is indecomposable.ppGq P ě Q K Q

G G G Now, let vtx and let Syl K such| K ÓQ thatÒQ “ K ÒQ. Since is -projective,

G G G P K ÓP | K ÒQÓP “ K ÒPX gQ gPrPzG{Qs so à P K ÒPX gQ g P G

by the Mackey formula, and hence, by the Krull-SchmidtG P Theorem, is a direct summand of for some , which is indecomposableK “ byK theÓP Claim.“ K ÒPX ThusgQ P X gQ “ P gQ “ P Q p G pK q “ ppGq KQ andKQ hence , so . Therefore, is a Sylow -subgroup of and it follows from Theorem 26.1(a), that vtx Syl . Finally, it is clear that the trivial -module is a -source, and hence all sources are trivial. 27 The Green correspondence

KG KL LTheď G Green correspondence is a correspondence which relates the indecomposable -modules with a ﬁxed vertex with the indecomposable -modules with the same vertex for well-chosen subgroups . It is used to reduce questions about indecomposable modules to a situation where a vertex of the givenp indecomposable module is a normal subgroup. This technique is very useful in many situations. In fact, many properties in modular representation theory are believed to be determined by normalisers ofLemma-subgroups. 27.1 Q ď G p L ď G U KG Q L ě Q Let be a -subgroup and let G . U ÓL Q (a) If is an indecomposable -module with vertex and , then there exists an inde- L ě NGpQq composable direct summand of with vertex . G V KL Q U V ÒL (b) If , then theG following assertions hold. V | U ÓL Q U (i) If V is an indecomposable KL-module with vertexQ and is a direct summand of G such that , then isU alsoV aÒ vertexL of . Q Q V Proof : (ii) If is an indecomposable -module which is -projective and there exists an indecomposable direct summand of with vertex , then is also a vertex of . Exercise, Sheet 5. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Theorem 27.2 (Green Correspondence) 75

Q p G L G NGpQq

Let beU a -subgroup of andKG let be a subgroup of Q containing . G (a) If is an indecomposable -moduleU withÓL “ vertexfpUq ‘ ,X then G fpUq U ÓL Q X L X xQ x P GzL where is the unique indecomposable direct summand of with vertex and every V KL Q direct summand of is -projective for some . G (b) If is an indecomposable -moduleV withÒL “ vertexgpV q ‘,Y then G gpV q V ÒL Q Y Q X xQ x P GzL where is unique indecomposable direct summand of with vertex and every direct gpfpUqq – U fpgpV qq – V summand of is -projective for some . f g (c) With the notation of (a) and (b), we then have and . In other words, „ and deﬁne a bijectionKG Q ÐÑ KL Q !isomorphism classes of indecomposable) !isomorphism classes of indecomposable) -modulesU with vertex ÞÑ fpUq -modules with vertex

gpV q Ð V [ .

TerminologyMoreover,fpU correspondingq modulesKL Green have correspondent a source in common.U Green correspondent gpV q KG Green correspondent of V Green correspondent V : is called the - of (or simply the ) Warning! Q and is called the - (or simply the of ). G Q L Q In the Green correspondence it is essential that a vertex is ﬁxed and not considered up to conjugation, because the -conjugacy class of and the -conjugacy class of do not coincide in general.Proof : Q X xQ L X xQ x P GzL x p˚q NGpQq ď L x Q Q X Q Q We ﬁrst note some properties of the subgroups and for : p˚˚q L X xQ Q L X xQ “ xQ p˚ ˚ ˚q Since L X, xQdoes not normaliseQ Land hence zisP aL proper subgroupL X xQ of “ . zQ x z z´ x ´ Q “ Qmay be ofQ the“ sameQ order as z , inx P whichNGpQ caseq ď L . x P zL “ L 1 x Supposex P GzL that L Xis conjugateQ to 1 in , i.e.Q thereL exists such that . Then V so andKL hence Q . Therefore . This contradicts . Therefore is never conjugate to in . G G Claim. KL V ÒL ÓL Q (b) Let be an indecomposable -module with vertex . L X xQ x P GzL L Pf of theThe Claim-moduleT KQhas a unique directV summand with vertexT Ò,Q and“ V all‘ otherZ direct summands are projective relative to subgroups of the form1 1 with .G G 1 KL Z KL V Z V ÒL ÓL “ V ‘ V G G 1 Z ÒL ÓL “ Z ‘ Z : Let be a -source for . Then, we may write for some -module . Moreover, there exist -modules and such that and T ÒG ÓG “ pV ‘ ZqÒGÓG “ V ÒGÓG ‘ Z ÒGÓG “ V ‘ V 1 ‘ Z ‘ Z 1 : . Then,Q L on the one handL L we haveL L L L Skript zur Vorlesung: Modular Representation Theory WS 2020/21

G G x xQ L T Ò Ó – p T Ó x qÒ x On the other hand, by the MackeyQ L formula we alsoLX haveQ LX Q xPrLzG{Qs à L x xQ L “ T ÒQ ‘ p T ÓLXxQqÒLXxQ xPrLzG{Qs àxRL x xQ L “ V ‘ Z ‘ p T ÓLXxQqÒLXxQ : xPrLzG{Qs àxRL

1 1 x xQ L V ‘ Z – p T Ó x qÒ x Therefore, by the Krull-Schmidt Theorem, we have LX Q LX Q xPrLzG{Qs àxRL L X xQ x R L V G G 1 1 V ÒL ÓL “ V ‘ V Q V x where, clearly, all direct summands are -projectiveL X Q forx someR L . It followsQ that is the uniquep˚ ˚ ˚q direct summand of with vertex , because all the direct summands in are projective relativeG to subgroups of the form with and so are not -projective by V ÒL KG U . This provesG the Claim. V | U ÓL Q V Q U G 1 Now,V ÒL write as a direct sum of indecomposableQ -modules and pick a directU summand G such that . By Lemma 27.1(b),V ÒL since is a vertex of , is also a vertex of . Therefore has at least one direct summand withV ÒG vertex“ U ‘ .U To1 ‘ proveX its uniqueness, assume is another indecomposable direct summand of . ThenL KG X

V ‘ V 1 “ U ÓG ‘U1 ÓG ‘X ÓG : for some -module , so in the notation of theL claim,L L G 1 G 1 V | U ÓL U ÓL | V 1 G y L U ÓL L X Q y R L V | T ÒQ 1 G As U | V ÒL, by the Krull-Schmidt-Theorem, and hence every indecomposable direct 1 L G G summand of is -projective forU some| T ÒQÒL “ byT Ò theQ : proof of the Claim. Now since and U1 Q it follows that Q1 Q 1 1 1 1 G Q ň Q S KQ U S | U ÓQ1 1 1 G 1 G L L Thus is -projectiveQ andď L thereforeU ÓQ1 “ hasU aÓL vertexÓQ1 containedS in . Y ÓQ1 G 1 It remains to prove that . SoY let beÓL a -source of . Then Q by The- 1 G y orem 26.1Y(b). Since , and henceU ÓL is a directL X summandQ of for y 1 ysomeR L indecomposable direct summand LofX U’Q . It followsy R L from Lemma L27.1 that isQ also a z y z´ y vertexQ1 Ď L ofX .Q But the indecomposablez P L Q direct1 Ď summandsQ ofz´ y R Lare all -projectivex P G forzL some 1 x 1 Q.Ď ThereforeQ X Q ň oneQ of thep˚q subgroups withgpV 1q “containsU an -conjugate of . I.e. for someU . Hence KG where andQ so there existsT KQ such G L G thatU U by . Therefore,T ÒQ“ weT mayÒQÒL set : and the assertion follows. L G (a) SupposeV T ÒQ now that Uis| anV indecomposableÒL -moduleV Q with vertex and letQ be a -sourceV of . Then is a direct summand of , so there is an indecomposable direct summand G of such that . This means that is -projective, and soY UisÓ aL vertex of byQ G G G G G LemmaU Ó27.1L | V(b).ÒL ÓL V ÒL ÓL Q G x Now,V by Lemma Y27.1– (a),V there exists an indecomposable directU summandÓL L X ofQ with vertex . Butx P GzL and the Claim says that the only direct summand of with vertex G G G G is . Therefore and the remaining direct summands of areU | U ÓL-projectiveÒL V | forV Ò someL ÓL . This proves part (a). (c) The claim follows immediately from parts (a) and (b) and the facts that and . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Exercise 27.3 77

(a) Verify that modules corresponding to each other via the Green correspondence have a source in common.

(b) Prove that the Green correspondent of the trivial module is the trivial module. 28 p-permutation modules

G X K ρX G KG ρX X K Recall from Example4(b)KX that any finite -setpermutationgives riseKG to a-module-representation on X of . The - module corresponding to through Proposition 10.3 admits as -basis, hence it is standard to denote this module by and it is called the . Thus, we may pose the followingDefinition definition. 28.1 (Permutation module) KG permutation KG K X G KX A -module is called a -module if it admits a -basis which is invariant under X G the action of . We denote this module by .

(Note: it is clearKG that the basis is then a ﬁnite -set.)

Permutation -modules and, in particular, their indecomposable direct summands have remarkable properties,Remark 28.2 which we investigate in this section.

KX KG X X First, we describe permutationn modules and some of their properties more precisely. G X “ i“ Xi KX KG If is a permutation -module on , then a decomposition of the basis as a disjoint union Ů n of -orbits, say 1 , yields a direct sum decomposition of as a -module as KX “ KXi : i“ à 1 X G

G Thus, without loss of generality, we can assumeKX – thatK ÒH is a transitive -set, in which case

H “ Gpxq G x P X K where : Stab , the stabiliser inKX of“ some K gx. Indeed, clearly we have a direct sum gPrG{Hs decomposition as a -vector space à G G KX “ K ÒH KG KG and acts transitivelyG on the summands, so that . K ÒH H ď G It follows that an arbitrary permutation -module is isomorphicG to a direct sum of -modules of K ÒH H ď G the form for various G . KG K ÒH “ KG bKH K “ gPrG{Hs g b K K Notice that, conversely, an induced module of the form ( ) is always a permutation À -module. Indeed, as as -vector space, it has on obvious Skript zur Vorlesung: Modular Representation Theory WS 2020/21

G K 78

-invariant -basis given by the set tg b K | g P rG{Hsu : H G KX KH X KX G ď 1 ÒH KG G K tg b x | g P rG{Hs; x P Xu In fact, more generally if and is a permutation -module on , then is a permutation -module with -invariant -basis . In other words, Exerciseinduction 28.3 preserves permutation modules.

Prove that direct sums, restriction, inﬂation and conjugation also preserveKG permutation modules.

Next, we investigate the indecomposable direct summands of the permutation -modules. In order to understand the indecomposable ones, we are going to prove that they all have a trivial source and we will apply the Green correspondence to see that, up to isomorphism, there are only a finite number of them.Definition 28.4 (trivial source module) KG trivial source KG KG K WarningA -module is called a -module if it is a finite direct sum of -modules with a trivial source . : Some texts (books/articles/. . . ) require that a trivial source module is indecomposable, others doProposition-Definition not. 28.5 (p-permutation module)

M KG P P ppGq

M G KQ p Q G Let beÓQ a -module and let Syl . Then, the followingď conditions are equivalent: G (a) M ÓP is a permutation KP-module for each -subgroup ; M K P (b) is a permutation -module; M KG (c) has a -basis which is invariant under the action of ; M KG (d) is isomorphic to a direct summand of a permutation -module; M p-permutation KG (e) is a trivial source -module. Note. p-permutation KG trivial source KG If fulﬁls one of these equivalent conditions, then it is called a -module.

Inp-permutation fact module -modules and trivial source-modules module are two diﬀerent pieces of terminology for the same concept. French/German speaking authors tend to favour the use of the termi- nologytrivial source module (and reserve the terminology for an indecomposable module with a trivial source), whereas English speaking authors tend to favour the use of the terminol- Proofogy : . ô g P G G g G M Ó gP – pM ÓP q G G (a) (b): It is obvious that (a) impliesM ÓP (b). For the suﬃcientKP condition, notice that forM eachÓQ , we have . Therefore, as by Exercise 28.3 restriction and conjugation preserve permutation modules, requiring that is a permutation -module implies that is a permutation Skript zur Vorlesung: Modular Representation Theory WS 2020/21

79 KQ p Q ď G p Q G p G ô -module for each -subgroup . (BecauseKP any -subgroup of is contained in a Sylow ñ Claim-subgroupL andKG these are all -conjugate by the Sylow theorems.) L (b) (c): is obvious by the definition of a permutationG -module. LÓP KP n P Zně (b) (e): : If is aQi ď-moduleG ď satisfyingi ď n (b), then so does any direct summand of . 1 Proof of the Claim: By Remark 28.2, if is a permutationn -module, then there exist G P and subgroups (1 ) such thatLÓP – K ÒQi ; i“ à P K Ò 1 Qi G N | L N ÓP where each KPis indecomposable by the Claim inN Example 11. Therefore, by the Krull–Schmidt theorem,M if , then is isomorphic to the direct sum of some ofM the factors, hence is againQ a permutation -module (by RemarkG G 28.2) and so satisfies (b)G as well, as required. M M | M ÓQÒQ Q M ÓQ KQ Now, if ô satisfies (b), then by the Claim we can assumen P Zě w.l.o.g. that isR indecomposable.i ď Q ď i ď n Let be a vertex of . Then by -projectivity. Since is a permutation -module n 0 by (a) ( (b)), again by Remark 28.2, thereM ÓG exist– KÒQ : and subgroups (1 ) such Q Ri that i“ à G M 1 K ÒG ď i ď n Ri Inducing thisQ module to againRi and“ Q using the Krull-SchmidtKQ theorem,K we deduce that , beingM indecomposable,M is isomorphic toKG a direct summand of for some 1 . By minimality ñ ofL the vertex , it follows that and that the trivial -moduleQ ď G must be a source of , G G provingL that| K ÒQis a trivial source -module.K ÒQ KG (e) (d): If is an indecomposable trivialKG source module, say with vertex , then by definition of a G G G ñ source, M |.Z This impliesZ (d) as is aKG permutation -moduleM ÓP | Z byÓP Remark Z28.2ÓP and any finite direct sumKP of permutation -module is again permutation. ñ G (d) (b): Assume that , whereM ÓPis a permutation KP-module. Then , where is again a permutation -module by Exercise 28.3. Thus, it follows from the Claim in (b) (e) (see also the scholiumñ below) that is a permutation -module, as required.

TheScholium Claim 28.6 in (b) (e) can be formulated as the following result. M p KG M p KG G p KG If is a -permutationKG -module, then any directp summand of is again a -permutation - module. In particular, if is a -group, then any direct summand of a permutation -module is a Exercise 28.7 permutation -module and so, in this case, any -permutation module is a permutation module. p

Prove that -permutation modules are preserved by the following operations: direct sums, tensor Exampleproducts, 12 restriction, inﬂation, conjugation, induction. KG p KG KG KG t u It is clear that any projective -module is a -permutation -module. Also, the PIMs of are p KG precisely the -modules with vertex 1 and trivial source. Q ď G Generalising this example, we can characterise the indecomposable -permutation -modules with a given vertex as described below. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Example 13 80

M p KG Q ď G Q KNGpQq fpMq M fpMq (1) IfK rNGispQ anq{Q indecomposables -permutationfpMq -module with vertex , then acts triv-

ially on the -Green correspondent of . Thus can beN viewedG pQq as a N K rNGpQq{Qs pNq -module. As such, is indecomposable and projective. NG pQq{Q KNGpQq Q KG (2) Conversely, if is a projective indecomposableKG -module,Q then Inf is an indecomposable p -moduleKG with vertex and trivial source. Its -Green correspondent is then also an indecomposable -module with vertex and trivial source, hence is an indecomposable -permutation -module

(3) In this way we obtain a bijection ÐÑ„ : p KG Q K rNG pQq{Qs ! isomorphism classes of indecomposable ) ! isomorphism classes of projective ) -permutation -modules with vertex indecomposable -modules

29 Green’s indecomposability theorem

To ﬁnish our analysis of indecomposable -modules we mention without proof an important indecomposability criterion due to J. A. Green (1959). The proof is rather involved and goes beyond the scope ofTheorem the techniques 29.1 (Green’s we have indecomposability developed so far. criterion, 1959) K H ď G G G p M KH M ÒH KGAssume that is an algebraically closed ﬁeld. Let be a subnormal subgroup of of index a power of and let be an indecomposable -module. Then is an indecomposable Proof : -module.

RemarkWithout 29.2 proof in this lecture. See [Thé95, (23.6) Corollary]. p indecomposability absolute indecomposability KG M Green’s indecomposability criterion remains true over an arbitraryKGpMq fieldsplit of characteristic local algebra , provided we replaceKGpMq{Jp KGpMqq – K with . (A -module is called abso- lutely indecomposable iff its endomorphism algebra End is a , that is, if CorollaryEnd 29.3 End .) K P p Q ď P M P KQ M ÒQ KP Assume that is an algebraically closed field. If is a -group, and is an indecomposable Proof : -module, then is anP indecomposablep -module.Q ď P p P By the Sylow theory, since is a -group, any subgroup can be plugged in a subnormal series where each quotient is cyclic of order , hence is a subnormal subgroup of . Therefore, the claim follows immediately from Green’s indecomposability criterion. M “ K KQ Notice that, in Example 11, we have proved the latter result in the particular case that is the trivial -module using simple arguments. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Exercise 29.4 81 K M KG

Q P pMq P P pGq Q ď P |P Q| pMq Assume that is an algebraically closedp ﬁeld and let be an indecomposable K-module. pMq p pMq “ pGq ˇ (a) Let vtx andK let Syl such that . Provep that : ˇ dim .

(b) Prove that if dim is coprime to , then vtx Syl . Chapter 8. p-Modular Systems

R. Brauer started in the late 1920’s a systematic investigation of group representationspF; O; kq overp-modular fields of systempositive characteristic. In order to relate group representationsO over fields of positivek “ O characteristic{JpOq to character theoryp in characteristic zero,F “ BrauerpO workedq with a triple of rings , called a , and consisting of a complete discrete valuationp ring with a residueBrauer field ’s reciprocity: theoremof prime characteristic and fraction field : Frac of characteristickG zero. The present chapter contains a short introduction to these concepts. WeFG will use -modular systems and in the subsequent chaptersOG to gain information about and its modules (which is/are extremelyK complicated) from the group algebra , which is semisimple and therefore much better understood, Notationvia the group algebra . This explains why we considered arbitrary associativep rings (resp. fields) G in the previous chapters rather than immediately focusingfinitely on fields generated of positive characteristic.G P t.F; ThroughoutO; ku this chapter, unless otherwise specified, we let be a prime number and denote a finite group. All modules are assumed to be and free as Λ -modules for References:any Λ .

Methods of representation theory. Vol. I

[CR90] C. W. Curtis andThe I. Reiner. block theory of finite group algebras. Vol. I . John Wiley & Sons, Inc., New York, 1990. [Lin18] M. Linckelmann. Representations of finite groups . Vol. 91. London Mathemat- ical Society Student Texts. Cambridge University Press, Cambridge, 2018. [NT89] H. Nagao and Y. Tsushima.Corps locaux . Academic Press, Inc., Boston, MA, 1989. [Ser68] Jean-Pierre Serre.G-algebras and modular. Deuxième representation édition, Publications theory de l’Université de Nancago, No. VIII. Hermann, Paris, 1968. [Thé95] J. Thévenaz.A course in finite group representation theory . Oxford Mathematical Mono- graphs. The Clarendon Press, Oxford University Press, New York, 1995. [Web16] P. Webb. . Vol. 161. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016. 30 Complete discrete valuation rings

In this section we review some number-theoretic results without formal proofs. I refer you to your number theory lectures for the details, or to [Ser68; Lin18; Thé95]. Important for the sequel is to keep the deﬁnitions and the examples in mind. 82 Skript zur Vorlesung: Modular Representation Theory WS 2020/21

83 O OzOˆ “ JpOq JpOq O ToO begin{JpOq with, recall from Chapter 1, §5, that a commutativek “ O{JpO ringq is local iff the residue, i.e. field of isthe the local unique ring maximalO ideal of . Moreover, by the commutativityp “ assumptionJpOq this happens if and only if is a field. In such a situation, we write : and callp this field . To ease up notation, we will often write : . This is because our aim is a situationDefinition in 30.1 which (Reduction the residue modulo field isp) a field of positive characteristic . O p “ JpOq k “ O{p M;N O f M ÝÑ N O Let be a local commutative ring with unique maximal ideal : and residue field : . k M “ M{pM – k b M reduction modulo p M Let be finitely generated -modules,O and let : be an -linear map. k f M ÝÑ N; m`pM ÞÑ fpmq`pN reduction modulo p (a) The -module : is called the of . f (b) The induced -linear map : is called the Exercise 30.2of . O p “ JpOq k “ O{JpOq

M;N O Let be a local commutative ring with unique maximal ideal : and residue ﬁeld : . f M ÝÑ N O f M ÝÑ N p (a) Let be ﬁnitely generated free -modules. f f

(i) Let : be anx ; :-linear : : ; xn P mapM n andP Zě : its reduction modulox ;:::.x ProveP M that if k is surjectiveM (resp.t anx ; :isomorphism), : : ; xnu O then is surjectiveM (resp. an isomorphism). 1 1 1 2 (ii) Prove that if elementsk pMq “ OpMq ( ) are such that their images 1 form a -basis of , then is an -basis of . O In particular, dim rk . M O Deduce that any direct summand of a ﬁnitely generated free -module is free. M (b) Prove that if is a ﬁnitely generated -module, then the following conditions are equivalent: M (i) is projective; O (ii) is free. M Moreover, if is also a PID, then (i) and (ii) are also equivalent to:

(iii) is torsion-free. Deﬁnition[Hint: Use 30.3 Nakayama’s Lemma.] O discrete valuation ring O JpOq ‰ A commutative ring is called a if is a local principal ideal domain such Examplethat 14 0. a p Zppq “ t b P Q | p - bu

Let be a prime number. We have already seen in Example1(b) that the ring : is commutative local with unique maximala ideal JpZ q “ t P Z | p | au “ pZ : ppq b ppq ppq Skript zur Vorlesung: Modular Representation Theory WS 2020/21

n 84 Zppq Zppq p Zppq n P Zě discrete valuation ring It follows easily that is a PID (every non-zero ideal in is of the form for some 0 integer ), hence a . O “ Rp R In fact this example is aR specialp case of a moreR general construction principlep forR discrete valuation rings, which consists in taking : , where is the ring of algebraic integers of an algebraic Remarknumber ﬁeld 30.4 and where is the localisation of at a non-zero prime ideal in .

discrete valuation ring ThereTheorem is in 30.5 fact a link between Deﬁnition 30.3 and the theory of valuations explaining the terminology O , provided by theπ followingP O result.Jp (NotOq “ diﬃcultπO to prove!)

a P Ozt u νpaq Let be a discrete valuation ring and let such that . Then: a P πνpaqO (a) For every 0 there is a unique maximal (non-negative) integer such that a; b P Ozt u νpabq “ νpaq ` νpbq νpa ` bq ě tνpaq; νpbqu . O πnO n P Z (b) For any 0 we have and min ě . ν Ozt u ÝÑ Z valuation 0 (c) Every non-zero ideal in is of the form for some unique integer . O The map : 0 defined in this way is called the of the discrete valuation valuation rings F ring . ν F ˆ ÝÑ Z One can use valuations to give an alternative definition of . Suppose that is a ¨ νpabq “ νpaq ` νpbq ν ñ νp q “ νpa´ q “ ´νpaq field and that : is a surjective map satisfying ¨ νpa ` bq ě tνpaq; νpbqu 1 (so is a group homomorphism 1 0 and ), and a; b P F νp q “ 8 min ,

for all , and we set for notationalO “ convenience ta P F | νpaq ě0 u . Then, the set

F “ : pOq 0 O O ˆ is a discrete valuation ring, andJpOq “Frac ta P F is| ν thepaq fraction ě u “ fieldOzO of : . Clearly, the unique maximal ideal in πis O νpπq “ O 1 Taking for any element in such that 1, one easily checks that has the properties stated in the theorem above. A valuation induces a metric, and hence a topology. For the purpose of representation theory of finite groups, we will need to focus on the situation in which this topology is complete. This can be expressed algebraicallyDefinition 30.6 as ( follows.Complete discrete valuation ring) O p “ JpOq

pamqmě O Cauchy sequence b ě b Let be a discrete valuationN ringě with uniquea maximalm ´ an P idealJpOq : m;. n ě N 1 (a) A sequence of elements of is called a if for every integer 1, there exists an integer 1 such that for all . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

85 O complete pamqmě Ď O a P O b b ě N ě a ´ am P JpOq m ě N 1 (b) is calleda limitif for every Cauchy sequencepamqmě there is Osuchcomplete that for withany integer respect to the1 therep-adic exists topology an integer 1 such that for all . (In 1 this case, is a of the Cauchy sequence .) It is also said that is Remark 30.7 . O A A JpAq O JpOq The previous deﬁnition can be generalised to a ﬁnitely generated -algebra . Moreover, one can Exampleprove 15that is complete in the -adic topology if is complete in the -adic topology. p Zppq p

Let be a prime number. The discrete8 valuation ring is not complete. However, its completion, the ring of -adic integers, that is, i Zppq “ Zp “ aip | ai P t ; : : : ; p ´ u @ i ě ; #i“ + ÿ ˆ 0 1 0 0 pZpq “ Qp p JpZpq “ pZp Zp{pZp – Fp is a complete discrete valuation ring. Its field of fractions is Frac , i.e. the field of -adic integers, and the residue field is .

FinallyCorollary we 30.8 mention without proof a very useful consequence of Hensel’s Lemma. O p “ JpOq k “ O{p p m P Zě p m Let ζbeP ak complete discrete valuationm ring with uniqueµ P O maximal idealµ “:ζ and residue ﬁeld 1 : of prime characteristic , and let be coprime to . Then, for any -th root of unity , there exists a unique -th root of unity such that . 31 Splitting p-modular systems

p-modular system In order to relate group representations over a field of positive characteristic to character theory in characterDefinition zero, 31.1 Brauer (p-modular chose systems a ) , which is defined as follows: p

pF; O; kq p-modular system Let be a prime number. O (a) A triple of rings is called a if: F “ pOq O (1) is a complete discrete valuation ring of characteristic zero, k “ O{JpOq O p (2) Frac is the field of fractions of (also of characteristic zero), and G p pF; O; kq splitting p-modular (3) is the residue field of and has characteristic . system for G F k G (b) If is a finite group, then a -modular system is a called a , if both and are splitting fields for . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

86 p

It is often helpful to visualise -modular systems and the condition on the characteristic of the rings Q Z F involved through the following commutative diagram of ringsp and ring homomorphisms:

F O k

where the hook arrows are the canonical inclusions and the two-head arrows the quotient morphisms. FG OG kG Clearly, these morphisms also extend naturally to ring homomorphisms g P G

Examplebetween the 16 corresponding group algebras (each mapping an element to itself). p G p One usually works with a splitting -modular system form all subgroups of , becauseQp it allowsm us avoid problemsG with ﬁeld extensions. By a theorem of Brauer on splitting ﬁelds such a -modular system can always be obtained by adjoining a primitive -th root of unity to , where is the exponent of . (Notice that this extension is unique.) So we may as well assume that our situation is as given in the following commutativeQp diagram:Zp Fp

F O k

More generally, we have the following result, which we mention without proof. The proof can be found inTheorem [CR90, 31.2 §17A]. pF; O; kq p G m

Let be a -modular system. Let be a finite group of exponent . Then the following F m F assertions hold. m (a) The field contains all -th roots of unity if and only if contains the cyclotomic field of F m k F k G -th roots of unity;

(b) If contains all -th roots of unity, then so does and and are splitting ﬁelds for Remark 31.3and all its subgroups. pF; O; kq p F k O If is a -modular system, then it is not possible to have and algebraically closed, while assuming is complete. (Depending on your knowledge on valuation rings, you can try to p prove this as an exercise!)

Let us now investigate changes of the coeﬃcients given in the setting of a -modular system for group algebras involved. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Definition 31.4 87 O OG L OG-lattice O Let be a commutative local ring. A finitely generated -module is called an if it Remarkis free 31.5 (= projective)(Changes of as the an coefficients-module. ) pF; O; kq p p “ JpOq L OG

¨ LF “ F b L FG Let be a -modularO system and write : . If is an -module, then: ¨ p L L “ L{pL – k b L kG setting : defines an -module, andO O OG L reduction modulo of , that is : defines a -module. F OG We note that, when seen as an -module, an -module may have torsion, which is lost on passage to . In order to avoidFG thismod issue, weO usuallyG lat only workkG mod with -lattices. In this way, we obtain functors OG FG kG - - - between the corresponding categoriesFG of finitely generated kG -lattices and finitelyO generatedG -, -modules.F A natural question to ask is: which -modules, respectively -modules, come from -lattices? In theProposition-Definition case of G-modules 31.6 we have the following answer. O F “ pOq O FG V OG L O F Let be aF complete discrete valuationV – ringL and let : LFracO-formbe theV fraction field of . Then, for any finitely generated -module there exists an -lattice which has an -basis which Proof : F tv ; : : : ; v u V L “ OGv ` ¨ ¨ ¨ ` OGv Ď V is also an -basis. In this situationn and we call an n of . L 1 1 Choose an -basis of and set : . Exercise: verify that is as required. kG

DeﬁnitionOn the other 31.7 hand, (liftable the questionkG-module has) a negative answer for -modules. O p “ JpOq k “ O{p kG M liftable OG M p M Let be a commutativeM local ring with unique maximal ideal : and residue ﬁeld : . A -module is called if there existsM{pM an– M:-lattice p whose reduction modulo of is isomorphic to , that is M liftable to an OG-lattice liftable to O liftable to p p characteristic zero (Alternatively, it is also said that is , or , or OG p kG kG .) OG Even though every kG-lattice can be reduced modulo to produce a -module, not every -module is liftable to an -lattice.kG Being liftable for a -module is a rather rare property. However, the next section brings us some answers towards classes of -modules which are liftable. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

32 Lifting idempotents 88

O p “ JpOq k “ O{p Throughout this section, we assume that is a complete discrete valuation ring with unique maximal ideal : andA residue field : . O O A O A V Moreover,pV ‰ V wepV let“ denotem a finitely generated -algebraA ofA finite{m -rank. ObserveA that since is a finitelym Ě pA generated -module, so is any simple -module and it follows from Nakayama’s lemma that , so 0. If is a maximal left idealpA Ď ofJpAq,: then is a simple -module and therefore . This provesJpAq that A JpAq k A “ A{pA p A It follows that is the inverse image in under the quotient morphism of the Jacobson radical of the finite dimensional -algebra : A{JpA(theq – reductionA{JpAq : modulo of ). Consequently, e R e ‰ e “ f ` g f g f “ g “ Recall that an idempotent of a ring is called primitive if 0 and whenever where Exerciseand are 32.1 orthogonal idempotents, then either 0 or 0. A e P A

JpeAeq “ eJpAqe Let be a ring and let be an idempotent element. Prove that: A k k e eAe (a) ; eJpAqe eAe (b) If is a ﬁnite-dimensional -algebra over a ﬁeld , then is primitive if and only if is a local ring. In that case is the unique maximal ideal of .

This leads us to the following crucial result for representation theory of finite groups on the lifting of idempotents.Theorem 32.2 (Lifting theorem of idempotents (partial version)) A O O A “ A{JpAq a P A a “ a ` JpAq Let be a finitely generated -algebra of finite -rank. Set : and for write ˆ : a P A . The followinga P Aˆ assertions hold.

(a) If , then . Thus there is a s.e.s. of groups ˆ ` JpAq Aˆ A :

1 1 1 x P A e P A e “ x

(b) For any idempotente; f P A , there exists an idempotentA esuchf that . A ´ e “ u f u u u P Aˆ e “ ufu´ ˆ ´ (c) Two idempotentse “ f 1 are conjugateu P in p `ifJp andAqq onlyĎ A if and aree “ conjugateufu in . More1 precisely if , then lifts to an invertible element such that1 . In particular, if e P,A then there existsA 1 e such that A .

(d) An idempotent is primitive in if and only if is primitive in . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

89 A Ñ A PpAq A PpAq (e) TheA quotient morphism induces a bijection between the set of conjugacy classes of primitive idempotents of and the set of conjugacy classes of primitive idempotents Proof : of .a P A Aa ‰ A a P m m A m Ě JpAq A a P m a (a) If is not invertible, then . Then for some maximal left ideal of by Zorn’s lemma.a P SinceA a ,“ itsx image isb a“ maximala ´ a left ideal of and we have . Thus is not

invertible.A The claim about the s.e.s. follows2 immediately. 1 1 1 1 1 (b) Let suchan “ thatan´ ` bn´and´ letan´ :bn´ . Deﬁne bybn induction“ an ´ a twon sequences@ n ě : of elements of : n 2 n 1 1 an ” an1 p 1 JpAq q bn P JpAq : 2 and : n` n 2 n` n “ 2 n bn P JpAq pJpAq q Ď JpAq We now provean “ bya inductionn ` bn that mod , or in2 other words1 that 2. This is true1

for 1, and2 assuming that this holds for , we have n` (because ), and since awen` obtain” an ` anbn ´ anbn p JpAq q 2 “ an2 ` bn ` anbn 2´ pan ` bnqbn 1 1 2 4 mod n` ” an ` bn ´ anbn “ an` p JpAq q : 2 4 1 pb q pa q 1 JpAq n ně 2 n ně mod A JpAq panqně 1 1 It follows thate P A convergese ´ toe “0 andnÑ8 thatbn “ eis a Cauchy sequenceA in the -adic 1 topology. Since is complete2 in the -adic topology (see Remark 30.7), converges to some element ˜ . Clearly, ˜ ˜ lim e “ a “ 0x, ; so ˜ is an idempotent of . Moreover

1 ˜ ´ e f A e f A e “ ufu´ ñ e “ u f u as required. ˆ ´ u P A e “ u f u 1 u P Aˆ 1 (c) It is clear that and´ are conjugate in if and are conjugate in ( ˆ). f ufu e “ f v “ A ´ e ´1 f ` ef v P A Conversely, assume that1 there exists such that . Then, by (a), and so, replacing by wev can“ assumeA ´ e ´ f ` .ef Now“ letA ´ e:´ f1` ef “ A :2 . Then, by (a), because ev “ ef “ vf e “ evv ´ “ vfv ´ 1 2 1 2 1 e A e 1 1 eAe Moreover, andJpeAe itq follows “ eJpA thatqe “ JpAq X eAe , as required. (d) The idempotent is primitive in if and only if and 0 are the only idempotents of . Since by the previous exercise eAe{JpeAeq “ eAe “, weeAe have :

f eAe eAe f f P eAe e e e Now, if is a non-trivial idempotent of , then byf (b)P appliedeAe to thef algebra , the idempotent lifts to an idempotent . Thise proves that is primitive if isf primitive. Conversely if is noteA primitive,e there exists a non-trivial idempotent . Then is not conjugate (that is, not equal) to 0 nor to the unity element . Thus, it follows from (c) that is a non-trivial idempotent of , as required. (e) This follows immediately from (b), (c) and (d). A{b b P A A ﬁrst possible application of Theorem 32.2 bis“ apA generalisation of this theorem providing a lifting of idempotents from a quotient for an arbitrary ideal . (See [Thé95, (3.2) Theorem].) We state here the particular case of interest to us for . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Theorem 32.3 (Lifting theorem of idempotents for reduction modulo p) 90 A O O A “ A{pA a P A a “ a ` pA Let be a ﬁnitely generated -algebra of ﬁnite -rank. Set : and for write x P A e P A e “ x : . The following assertions hold. ˆ Aˆ “ ta P A | a P A u (a) For every idempotent , there exists an idempotent such that . e ; e P A e “ e u P Aˆ (b) . e “ ue u´ 1 2 1 2 (c) If 1 are idempotents such that then there is a unit such that 1 2 A A A . A (d) The quotient morphism induces a bijection between the central idempotents of and Proof : the central idempotents of .

Exercise! (Either use Theorem 32.2 or imitate the proof of Theorem 32.2.) kG OG The lifting of idempotents allows us, in particular, to prove that projective indecomposable -modules areLemma liftable 32.4 to projective indecomposable -lattices. A R P A e P A P – Ae Let be a ﬁnitely generated algebra over a commutative ring . If is a projective indecomposable ˝ n Proof : P P | pA q n P Zě P -module, then there exists an˝ idempotent˝ such that . P | A A “ P ‘ Q A Q A “ e ` f 1 Sincee P P isf projective,P Q for some . As is indecomposable, it follows from the Krull-Schmidt theorem that , sofe “ p ´ eqefor“ somee ´ e “-moduleef . Thus, we can write 1 with and . ThenAe X Af Ď Q X P “ t u 2 e “ e ef “ fe “ 1 2 A “ A ¨ “ Ape ` fq “ Ae ` Af ; Ae Ď P Af Ď Q; and this is an element of A 0 . Therefore and 0. Finally, since Ae “ P 1 and

Corollaryit follows 32.5 that . A O O A “ A{pA P A A P LetP – Pbe{pP a ﬁnitely generated -algebra of ﬁnite -rank and set : . Let be a projective (indecomposable) -module. Then there exists a projective (indecomposable) -module p such that Proof : p P p“. P ‘ ¨ ¨ ¨ ‘ Pr r P Zě P A f ; : : : ; fr P A Pi – Afi 1 1 ďLeti ď r ( ) be a decomposition of as a direct sum of indecomposable - 1 submodules. Then, by Lemma 32.4, thereP exist– Af idempotents‘ ¨ ¨ ¨ ‘ Afr : such that for each 1 , and so 1 e ; : : : ; er P A ei “ fi ď i ď r P “ Ae ‘ ¨ ¨ ¨ ‘ Aer A P{pP – P 1 Now, by TheoremP 32.3(a) there existsP idempotents such that for each 1 . 1 Then p : is a projective -module (see Example 10) and p p . Moreover,P “ P ‘pPis indecomposableñ if Pis.{pP Indeed,– k bO pP ‘ P q – pk bO P q ‘ pk bO P q :

1 2 1 2 1 2 p p p decomposable p p p p p p decomposable Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Corollary 32.6 91 kG OG

ProofAny : (projective) indecomposable -module is liftableA “ toOG a (projective) indecomposable -lattice. A “ OG{pOG – k b OG – pk b OqG – kG : This follows immediately from Corollary 32.5 Owith : O since then

Exercise 32.7 A O O M A e P A ApMq Let be a ﬁnitely generated -algebra of ﬁnite -rank. Let be an -module and let be an idempotent. Prove that, as End -modules,ApAe; Mq – eM :

Hom 33 Brauer Reciprocity

pF; O; kq p G p “ JpOq p “ JpOq Throughoutp ą this section,V we let FG be a splitting -modular systemO forL andV write : . Reduction modulo : p LprovidesL “ usL with{p a method for going from characteristicO zero to character- F F istic 0. Moreover,L ifL is an O-module, thenV we canpL chooseq – V an– p-formL q of and then consider theL ﬂ reductionL modulo of , i.e. . However, the choice of the -form is not unique. As a 1 2 1 2 consequence, if and are two -forms of , i.e. , then it may happen that 1 2 . The following result shows that, however, this does not aﬀect the composition factors, up to isomorphismProposition 33.1 and multiplicity.

S ;:::;St t P Zě kG V FG L O V ď j ď t 1 1 LetSj ( ) be a completeL “ L{p setL of representatives of the isomorphismO classesL of simple -modules. If is an -module and is an -form of , then for each 1 the multiplicity Proofof : asj aP compositiont ; : : : ; tu factork of does notG dependkGpS onj q the – k choice of the -form . Sj L Fix 1 . Since is a splitting ﬁeld for , End . Thus, by Proposition 22.1(b), pP ; Lq{ pS q “ pP ; Lq : the multiplicity of as ak compositionkG Sj factor ofk is kG j k kG Sj e P OG dim Hom dim End dim Hom j P – kGe : On the other hand, by Lemma 32.4 and TheoremSj 32.3, therej exists an idempotent such that

pP ; Lq – pkGe ; Lq – e L – e L: Hence, Exercise 32.7 yields kG Sj kG j j j

Hom Hom pP ; Lq “ pe Lq “ pe Lq “ pe V q : Then, Exercise 30.2(a)(ii) andk Proposition-DeﬁnitionkG Sj k j 31.6 yieldO j F j

dim Homď j ď t dim rk dim L Sj F pej V q O L As a consequence, for any 1 , the number of composition factors of isomorphic to is equal to dim , and is therefore independent of the choice of the -form . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Theorem 33.2 (Brauer Reciprocity) 92

V ;:::;Vl l P Zě FG S ;:::;St t P Zě 1 1 Let ( kG ) be a completee ; : set : : ; e oft P representativesOG of the isomorphismkGe classesj of simple 1 1 -modules,Sj and let ď j ď t ( ) beď ai completeď l setď j ofď representativest dij of the isomorphism 1 classesSj of simple -modules. Let bep idempotentsO suchVi that is a projective cover of for each 1 . For every 1 and 1 deﬁne to be the multiplicity l of as a composition factor of the reduction modulo of an -form of . Then FGej – dij Vi: i“ à 1

Proof : FG F tVi | ď i ď lu FG F GSince is a semisimpleVi -algebraď i ď l the set 1 F Visi a complete set of representatives 1 of the isomorphismFG classes of the PIMsď j ofď t (by Theorem 21.2(b)). Moreover, asdij is a splitting ﬁeld for , by Theorem 8.2, each (1 ) appears precisely dim times in the decomposition of the l regular module . Hence, for any 1 , there exist1 non-negative integers such that FGej “ dij Vi ; i“ à 1 1 dij “ F FGpFGej ;Viq 1 dij “ dij ď i ď l ď j ď t O Li Vi ď i ď where dim Hom . Thus, it remains to prove that for every 1 and

1 1 . So choose an -form of (1 ). As in the previous proof, applying Exercise 32.7 and Propositiondij “ 31.6F yieldsFGpFGej ;Viq “ F pej Viq “ Opej Liq “ k ej Li “ k kGpkGej ; Liq “ dij :

dim Hom dim rk dim dim Hom Deﬁnition 33.3 (Decomposition matrix)

dij ď i ď l; ď j ď t p decomposition numbers G The positive integers (1 1 ) deﬁned in Theorem 33.2 are called the - ppGq “ pdij q ďiďl of and the matrix ďjďt 1 p-decomposition matrix ofDecG : 1decomposition matrix G

Exerciseis called 33.4 the (or simply the ) of the ﬁnite group . C kG D “ ppGq G

tr Let be the Cartan matrix of and : DecC “ D Dbe the decomposition matrix of . Prove that C

and deduce again that is symmetric. Chapter 9. Brauer Characters

F FG FG Recall that if is a field of characteristic zero, then -modules are isomorphic if and only if their characters are equal. Also,k the character of ap ą -module providesW completek information about its compositionG factors, includingk pW q “ multiplicities,ap ` provided that the irreduciblea charactersk are known.W All this does not hold for fields of characteristick 0. For instance, if is a -vector space on which acts trivially and dim p 1 for some nonnegative integer , then the -character of is the trivial character.character This implies theory that a -character can only give information about multiplicities of composition factors modulo . In view of these issues, the aim of this chapter is to define a slightly different kind of for modular representations of finite groups and to establish links withNotation ordinary characterG theory. p pF; O; kq p F pGq pF; O; kq p : Throughout,G denotes a finite group and a prime number.p We“ Jp letOq P tdenoteF; O; k au -modularG system and we assume contains all exp -th roots of unity, so P tF;is ku a splittingU - modularG system for and allρU itsG subgroupsÝÑ pU (seeq Theorem 31.2). We write : . For Λ all Λ -modules considered are assumed to be free of finite rank over Λ. If Λ and is a Λ -module, then we write : GL for the associated Λ-representation.Character Theory of Finite Groups For background results in ordinary character theory I refer to my Skript References:from the SS 2020.

Local representation theory

[Alp86] J. L. Alperin. Methods of representation. Vol. 11. Cambridge theory. Vol. Studies I in Advanced Mathe- matics. Cambridge University Press, Cambridge, 1986. [CR90] C. W. Curtis andCharacter I. Reiner. theory of finite groups . John Wiley & Sons, Inc., New York, 1990.Character theory of finite groups [Isa94] I. M. Isaacs. . Dover Publications, Inc., New York, 1994. [Las20] C. Lassueur. Representations of. Lecture finite groups Notes SS 2020, TU Kaiserslautern, 2020. [NT89] H. Nagao andCharacters Y. Tsushima. and blocks of finite groups . Academic Press, Inc., Boston, MA, 1989. [Nav98] G. Navarro.A course in finite group representation. Cambridgetheory University Press, Cambridge, 1998. [Web16] P. Webb. . Vol. 161. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016.

93 Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Deﬁnition Before we start, we recall the following useful terminology from ﬁnite group theory. G p

a a |G| “ p m a; m P Zě pp; mq “ |G|p “ p p part |G| Let be a ﬁnite group and be a1 prime number. G |G|p1 “ m p part G 0 (a) If with and 1, then : is the - of the order g P G p-regular element p1 element p opgq of and : is the - of the order of . -

(b) An element is called a Gp1 “ tg P G |(orp - aopg-qu ) if and we write

p : G g P G p-singular element p | opgq p-element for the set of all -regular elements of . (Warning: in general this is not a subgroup!) opgq p (c) An element is called a if and it is called a if is a power of . Remark n G p g P G opgq “ p m n; m P Zě pp; mq “ a; b P Z “ apn ` bm 0 Let be a ﬁnite group and be a prime number.bm Let andapn write with such that 1. Then, lettinggp “ g be such thatgp1 1“ g , we may set

pn : mand : pgpq “ ; pgp1 q “ ; g “ gpgp1 : It is immediate that 1 gp p p-part g gp1 p p -part g 1 1 and Thus, is a -element and is called the of , and is -regular and is called the of . 34 Brauer characters

p pF; O; kq F pGq F k a a pSince we assume that the given -modular system is such that contains all exp F-th roots kof unity, both and contain a primitive -th root of unity, where is the l.c.m. of the orders of the -regular elements. To start with weµF examine“ ta the relationshipF betweenu the roots of unity in and in . Set µk “ ta ku : : -th roots of 1 in ; µF Ď O µF µk : -th roots of 1 in O O{p µF

Then and, as both and are ﬁnite groups,– it follows from Corollary 30.8 that the quotient morphism restricted to induces aµF groupÝÑ µ isomorphismk :

ξ ÞÑ ξ ξ a k ξ a O We write the underlying bijection as p , so that if is an -th root of unity in then p is the unique -th root of unity in which maps onto it. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Lemma 34.1 (Diagonalisation lemma) 95

ρ G ÝÑ pUq k G p g P Gp1 k ρpgq ρpgq opgq

Letµk : GL be a -representation ofk . Then,B forU every -regular element , the -linear map is diagonalisable and the eigenvalues of are -th roots of unity and lie in . In other words, there exists an ordered ξ-basis of with respect to which ξ » 1 fi ρpgq B “ 0 0 ; — 2 ffi ` ˘ — 0 ffi — ξ ffi — nffi – 0 fl n U ξ i n o g k “ k p q i ď ď 0 p q 0

Proofwhere : g :P Gdimp1 and each (1 ) is an ρ -th root of unity in .xgy p - |xgy| kxgy k xgy Let . It is enough to considerk the restrictionxg ofy to the cyclic subgroupρ|xgy . Since , is semisimple by Maschke’s Theorem. Moreover, as is a splittingρpgq “ ﬁeldρ|xg forypgq , it follows from Corollary 12.3 that allk irreducibleB U-representations of have degree 1. Hence can be decomposed opgq as the direct sum ofo degreepgq 1 subrepresentations. AsρU p ag consequenceq “ ρU p Gq “ U is diagonalisableµk andopgq there exists a -basis aof satisfying the statement of the lemma. It follows immediately that the eigenvalues are -th roots of unity because 1 Id . They all lie in , being -th roots of unity, hence -th roots of unity.

ThisDeﬁnition leads 34.2 to the (Brauer following characters deﬁnition.)

U kG n P Zě ρU G Ñ pUq k p-Brauer character Brauer character G U 1 LetρU be a F -module of dimension and let : GL be the associated - representation. The or simply the of aﬀorded by (resp. of ) is the -valued function 'U Gp1 Ñ O Ď F g ÞÑ ξ ` ¨ ¨ ¨ ` ξn ; : 1 ξ ; : : : ; ξn P µk ρU pgpq p n degree 'U 'U irreducible U ρU linear 1 wheren “ are theppG eigenvaluesq of . The integer is also calledG the of1Gp1. Moreover, is called if kGis simple (resp. if is irreducible), and it is called if 1. We denote by IBr the set of all irreduciblekG Brauer characters of and we write C for the Brauer character of the trivial -module. In the sequel, we want to prove that Brauer characters of -modules have properties similar to - characters.Remark 34.3

Warning 'pgq P O Ď F ρU pgq k p ą

(a) : even though is defined over the field of characteristic 0. 'U pgq (b) µOftenF theC values of Brauer characters are considered'U pgq as complex numbers, i.e. sums of complex rootsρU p ofgq unity. Of course, in that case then depends on the choice of embedding of into . However, for a fixed embedding, is uniquely determined up to similarity of . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

ImmediateProposition properties 34.4 of Brauer characters are as follows.

U ‰ kG 'U ' p q “ pUq Let U 0 be a k-module with Brauer character . Then:

' G 1 (a) U 1 dim ; p ´ ' pg q “ ' ˚ pgq @ g P G 1 (b) U is a classU function on p ; 1 (c) . Proof : n “ k pUq ρU k U ρ p q U n “ Let : dimU G and let be the -representation associated to . k k F n “ k pUq ´ ´ ´ p (a) Clearlyg; x P Gp11 is theρU identitypxgx q map “ ρ onU pxqρandU pgq hasρU pxqeigenvaluesρU pg allq equalρU p toxgx1 . Sinceq 1 1 the ´ sum of the lifts is dim 1 . 1 'U 'U pgq “1 'U pxgx q (b) If , then x P G , so and are similar1 and

thereforeg P G havep1 the sameξ ; : eigenvalues. : : ; ξn P µk Thus, by deﬁnition ofρU pgweq haveρU pgq . [Note that in this proof we could take and it would not change´ the result!]´ ´ ´ ρU pg q “ ρU pgq ξ ; : : : ; ξn 1 ´ tr (c) Let and let ρU˚ pgq “ beρU p theg eigenvaluesq µF ofÝÑ µk.;1ξ AsÞÑ ξ is1 diagonalisable1 by1 1 Lemma 34.1´ , it follows´ immediately that the eigenvalues of are , ξ “ ξi ď i ď n 1 i p as are the eigenvalues1 1 of . Since is a group isomorphism, y p Exercisewe 34.5 have for each 1 , and the claim follows. U;V;W kG

U V W kG Let be non-zero -modules. Prove the following assertions.

(a) If 0 0 is a s.e.s.' ofV “ -modules,'U ` 'W : then

U S ;:::;Sm m P Zě n ; : : : ; nm

1 1 1 (b) If the composition factors of are'U “ n 'S ` :::( ` nm'S)m with: multiplicities respectively, then kG 1 1

In particular, if two -modules have isomorphic composition factors, counting multiplicities, ' “ ' ` ' ' “ ' ¨ ' thenU‘V they haveU theV sameU Brauerbk V character.U V

(c) and . good enough The link between the Brauer characters and the usual notion of a character, i.e. trace functions, is Deﬁnitiondescribed below 34.6 ( andcharacter explains) why trace functions are not .

P tF; ku U ‰ G ρU G ÝÑ pUq

Let Λ . Let 0p´be;U aqΛ G-moduleÝÑ and let : GL be the associated Λ- representation, then the trace function g ÞÑ pg; Uq “ ρU pgq Tr : Λ ` ˘ Tr : Tr Skript zur Vorlesung: Modular Representation Theory WS 2020/21

-character U 97

“ F χU p´;Uq F pGq Fis called the Λ G of . For Λ , we also write instead of Tr and we write Irr for the set of all irreducible Character Theory of Finite Groups F χU U -charactersFG of . (Recall from that an -character is called irreducible if is a simpleLemma 34.7-module.) U ‰ kG g P G

pg; Uq “ pg 1 ;Uq Let 0 be a -modulep and let . Then: pg; Uq “ ' pg 1 q ` p k “ O{p (a) Tr TrU p , and

(b) Tr in the quotient . Proof : ρU k U

ρ pgq “ ρ pgpqρ pgp1 q “ ρ pgp1 qρ pgpq Let be theU -representationU U associatedU to U . ρU pgq ρU pgp1 q ρU pgpq (a) Clearly gp p p ρU pgpq . Therefore, we know from linear algebra thatρU pg thepq eigenvalues of are pairwisepkq product “ p of suitablyρU pg orderedq ρ eigenvaluesU pgp1 q of and . Now, as is a -element, a -power of is the identity and so all the eigenvalues of

are equal to one because char .1 Hence and have the same eigenvalues ξ ; : : : ; ξn ρU pgp q 'U pgq “ ξ ` ¨ ¨ ¨ ` ξn (counted with multiplicities) and the claim follows. ď i ď n ξi ξi ` p “ ξi k “ O{p 1 1 (b) Let be the eigenvalues of . Then, by deﬁnition, p p , where, for each 1 , p is such that p in the quotient . Therefore, it follows from (a) g; U g 1 ;U ξ ξ ξ p ξ p ξ ::: ξ p ' g 1 p : andp theq diagonalisation “ p p q “ lemma`¨ ¨ ¨` thatn “ p ` q`¨ ¨ ¨`p n ` q “ p ` ` nq` “ U p p q`

1 1 1 Tr Tr p p p p Notation 34.8

F pGq “ tf G ÝÑ F | f u F F G F pGp1 q “ tf Gp1 ÝÑ F | f u F WeF let Cl : : Gp1 class function denote the -vector space of -valued class functions on and we let Cl : : class function denote the -vector space Propositionof -valued 34.9 class functions on . ' P pGp1 q Z ppGq A class function Cl is a Brauer character if and only if it is a non-zero non-negative Proof-linear : combination of elements of IBr . n ; : : : ; nm P Zě ' ;:::;' P pGq pm P Z kG S ;:::;S S Sm p ě m 1 0 ' “ n ' ` ::: ` nm' 1 The necessary condition is clear by Exercise 34.5S (b). Conversely,Sm let and let kG 1 1 IBr ) be the Brauer characters1 aﬀorded by simple -modules . n 1 nm Then, by Exercise 34.5(c), the class functionS :‘ ::: ‘ Sm : is the Brauer character aﬀorded by the -module 1 1 Theorem 34.10

ppGq G F

| ppGq| ď F F pGp1 q “ p G: The set IBr of irreducible Brauer characters of is -linearly independent and hence ppGq “ t1G u G p IBr dim Cl p1 number of conjugacy classes of -regular elements in In particular, IBr provided is a ﬁnite -group. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

98 p Without proof. F G p Gp1 “ t Gu

|Thep secondpGq| “ claim is obvious, because thep indicatorpGq “ t1G functionsp1 u on the conjugacy classes of -regular elements form an -basis. The third claim is clear since if is a -group, then 1 . Thus IBr 1 by the ﬁrst part, namely IBr . 35 Back to decomposition matrices of ﬁnite groups

G F C G k F We now want to investigate the connections between representations of over (or ) and represen- tationsLemma of 35.1over through the connections between their -characters and Brauer characters.

V FG F χV L O V L “ L{pL p L χ | “ ' ; Let be an -module with -character V .Gp Let1 Lbe an -form of and let be the reduction modulo of . Then, reduction modulo p of χV

Proofand : is ofteng P G calledp1 the RL G ÝÑ pOq . O L g p x ; : : : ; xn RLpgq µF Ď O x ;:::; xn P k ρ g ' LetLp q and write L: GL for the -representation associated to . Since is 1 1 -regular, the eigenvalues of belong to and are the eigenvalues F of . Since by deﬁnition'Lpgq “isx the` ::: sum` ofxn the“ liftspg; ofL theq “ latterpg; V quantities,q “ χV pgq we have

L O V 1 Tr Tr

Propositionsince is 35.2 an -form of .

ppGq F F pGp1 q F χ P pGq d (a) The set IBr is an -basis of Cl F . χ' χ| “ d ': (b) Given an irreducible -character GIrrp1 , there existχ' non-negative integers such that ' G P ÿpp q IBr | ppGq| “ p G

(c) IBr number of -regular conjugacy classes of . Proof : ppGq F pGp1 q µ P F pGp1 q F Gp1 µ µ P F pGq µ pgByq “ Theorem 34.10g PitG onlyzGp1 remains toF p proveGq thatFIBr spansF pGClq # . So let Cl be an# arbitrary -valued class function on . Then can be extended to Cl , e.g. by setting 0 for every . Since Irr µ “is an -basisaχ ¨ χ for Cl , we can write χ G # P ÿF p q Irr aχ P F @ χ P F pGq

µ “ µ | “ a ¨ χ| : with Irr . Thus Gp1 χ Gp1 χ G # P ÿF p q Irr Skript zur Vorlesung: Modular Representation Theory WS 2020/21

99 χ| G χ| Gp1 Gp1 µ F where by Lemma 35.1 each is a Brauer characterF of , and hence bypp PropositionGq 34.9, each isF ap non-negativeGp1 q integer linear combination of irreducible Brauer characters. Therefore is an -linear combination of irreducible Brauer characters over . It follows that IBr is a generating set for ExerciseCl 35.3. This proves (a). Parts (b) and (c) are immediate consequences of (a) and its proof. kG

Prove that two -modules aﬀord the same Brauer character if and only if they have isomorphic Remarkcomposition 35.4 factors (including multiplicities). FG kG dχ' p If we translateG part (b) of Corollary 35.2 from irreducible characters and Brauer characters to - modules and -modules, we obtain that the integers are the same as the -decomposition D “ pGq “ pd q : numbers of as deﬁned through Brauer’s reciprocityp χ' theorem,χP F pG i.e.q 'P ppGq Irr G : Dec IBr

C “ DtrD “ pc q The Cartan matrix of is then 'µ ';µP ppGq

'; µ P ppGq IBr

(see Exercise 33.4) and for every c'µIBr“ we havedχ'dχµ: χ G P ÿF p q Irr Corollary 35.5

ppGq G | ppGq| G (a) The decomposition matrix Dec of has full rank, namely IBr .

(b) The Cartan matrix of is a symmetric positive deﬁnite matrix with non-negative integer

Proof : entries. tχ| | χ P F pGqu F pGp1 q F Gp1 B Ď tχ| | χ P F pGqu F F pGp1 q Gp1 It follows from Proposition 35.2 that Irr spans Cl over . There is therefore pdχ'qχPB;'P ppGq F ppGq a subset Irr which forms an -basis for Cl . Now by Proposition 35.2, the IBr columns of the matrix are -linearly independent, hence Dec has full rank. This proves (a) and (b) followskG immediately. F G kG Recall now that projective -modules are liftable by Corollary 32.5. This enables us to associate an Deﬁnition-character 35.6 of to each PIM of .

' P ppGq kG S PS F S PS PS O F pPSq Let IBr ' be an irreducibleprojective Brauer character indecomposable aﬀorded characterby a simple -module S. Let' be the projective cover of and let p denote a lift of to . Then, the -character of p is denoted by Φ and is called the associated to or . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Corollary 35.7 100

' P ppGq

' “ dχ'χ Let IBr χP. Then:F pGq

'| ř“ Irrµ G c'µµ (a) Φ Gp1 P pp q ; and ř (b) Φ IBr . Proof :

(a) This follows from Brauer reciprocity. '| “ dχ'χ| “ dχ' dχµµ “ c'µµ : (b) This follows fromG partp1 (a) because Gp1 χ G χ G µ G µ G P ÿF p q P ÿF p q P ÿpp q P ÿpp q Φ Irr Irr IBr IBr Theorem 35.8 p - |G| V FG L O V p Assume , then the following assertions hold: kG (a) If is a simple -module and is an -form for , then its reduction modulo is a simple -module and the map F pGq ÝÑ ppGq χ ÞÑ χ | “ χ V V Gp1 V Irr IBr

G pGq G is a bijection. p

(b) Both the Cartan matrix of and the decomposition matrix Dec of are the identity matrix, provided the rows and the columns are ordered in the same way. Proof : p - |G| Gp1 “ G kG kG kG S kG ' Since , clearly . Now, by Maschke’s theorem, is semisimple, so the simple - modules are precisely the PIMs of . Thus, if is a simple -module aﬀording the Brauer character , c “ δ @ µ P G: then by deﬁnition of the Cartan integers'µ (Def. 22.2'µ ) and Remarkp 35.4 we have

IBr “ c'' “ dχ' χ G Therefore, P ÿF p q 2 χ 1P F pGq Irr dχ ' ‰ dχ ' “

0 0 and it follows that there is a unique 0 Irr such that 0; in fact we must have 1. ' “ χ χ | “ '; Now, it follows from Corollary 35.7 that Gp1

0 0 Φ and f so the ﬁrst claim of (a) holds and the given map is well-deﬁned and surjective. It follows immediately | pGq| “ G G “ | pGq| ă 8 : that is bijective since byp Proposition 35.2(c) we have F

IBr # -conjugacy classes in Irr

This proves (a) and (b). Skript zur Vorlesung: Modular Representation Theory WS 2020/21

101

Finally, we would like to obtain orthogonality relations for Brauer characters, which generalise the row orthogonality relations for ordinary irreducible characters. However, unlike the case of ordinary characters, there are now two signiﬁcant tables that we can construct: the table of values of Brauer characters of simple modules, and the table of values of Brauer characters of indecomposable projective modules.Deﬁnition 35.9 (Brauer character table)

l “ |Gp1 | g ; : : : ; gl p G 1 Set : and let be a complete set of representatives of the -regular conjugacy Brauer character table G 'pgj q P MlpFq classes of . 'P ppGq ´ ¯ ďjďl (a) The of a finite group is the matrix IBr . Brauer projective table G p g1 M F 'p j q 'P ppGq P lp q ´ ¯ ďjďl IBr (b) The of a finite group at is the matrix Φ 1 . Definition 35.10

F f ; f G Gp1

1 2 ´ Given -valued class functions xf ; f deﬁnedy 1 “ on a subsetf pgq off pg containingq : , deﬁne p |G| gPG 1 p 1 1 2 1 ÿ 1 2 :

Note x ; yp1 F F pGp1 q

. It is straightforward that is -bilinear on Cl . p With these tools we obtain a replacement for the row orthogonality relations for ordinary irreducible characters. It says that the rows of the Brauer character table and of the -projective Brauer table are orthogonalTheorem 35.11 to each other.

' ' P ppGq p t ' | ' P ppGqu F

(a) All projective indecomposable˝ characters Φ ( IBr ) vanish on -singular elements, and the set Φ IBrF pGq “is t anf P -basisF pGq of| fp thegq “subspace@ g P GzGp1 u

F pGq Cl : Cl 0

'; ψ P pGq x'; y 1 “ δ “ x ; ψy 1 of Cl . p ψ p 'ψ ' p

(b) For every IBr we have Φ Φ . Proof : g ; : : : ; gr G g ; : : : ; gl p gl` ; : : : ; gr p l ă j ď r 1 1 Let be a complete set of representatives of the conjugacy classes of such that 1 ´ ´ ´ χpg q “ χ| pg q “ dχ''pg q are -singular and arej -regular.Gp1 Ifj , then by Propositionj 35.2(b), ' G 1 1 P ÿpp q 1 ď i ď r IBr F

and for each 1 the 2nd Orthogonality Relations for the irreducible -characters (see [Las20, Skript zur Vorlesung: Modular Representation Theory WS 2020/21

102

δ C g χ g χ g´ χ g d ' g´ Thm. 12.2])p˚q yield ij | Gp iq| “ p iq p j q “ p iq χ' p j q χ G χ G ' G P ÿF p q 1 P ÿF p q ´ P ÿpp q 1 ¯ ´ Irr “ Irr IBrdχ'χpgiq 'pgj q ' G χ G P ÿpp q ´ P ÿF p q ¯ 1 ´ “ IBr 'pgIrriq'pgj q ' G P ÿpp q 1 IBr ďΦi ď l pg q' “ 'P ppGq ' i pG 1 q pg q “ ' P pGq pGq F F p ' i p p ř where the last equality holds by Corollary 35.7(a). Then for each 1 , we have IBr Φ 0 ˚ in Cl and we obtain that Φ 0 for each trIBr since IBr is -linearly independent. Thus the ﬁrst claim of (a) is established. 'pgiq Now, ( ) says that the square matrix |CGpgiq| 'P ppGq ˆ ˙ l` ďiďr Φ IBr 1 ´ 'pgi q : is a left inverse for the square matrix 'P ppGq ˆ 1 ˙ l` ďiďr IBr 1

Therefore, it is also a right inverse,r and multiplying both matrices the other way around, we obtain ´ ´ (applying the Orbit-Stabiliserδ'ψ “ Theorem) that'pgi q ψ pgiq “ 'pg q ψ pgq |CGpgiq| |G| i“l` gPGp1 ÿ 1 1 1 ÿ 1 Φ Φ 1 “ 'pgq pg´ q |G| ψ gPGp1 1 ÿ 1 “ x'; ψ yp1 Φ

'; ψ P ppGq x '; ψy 1 “ δ'ψ '; ψ P ppGq p Φ t '| | ' P ppGqu ppGq F pGp1 q F x; y 1 Gp1 p for all IBr . Similarly Φ for all IBr . This means in eﬀect that Φ IBr and IBr are dual bases of Cl with respect to the -bilinear form . RemarkThus, 35.12 we have proved (a) and (b).

B “ p|CGpgl` q|;:::; |CGpgrq|q tr ´ The proof of the theoremB tells“ I us that writing Φ for the Brauer projective table, Π for the Brauer 1 Exercisecharacter 35.13 table and setting1 : diag , then the orthogonality relations can be written as Φ Π . 1 H p G k F H G ExerciseLet 35.14be a -subgroup of a ﬁnite group . Prove that the character Φ is a constituent of the trivial -character of induced to . '; λ P ppGq λ λ' P ppGq λ ' “ λ'

ExerciseLet 35.15IBr and assume that is linear. Prove that IBr and Φ Φ . G ρreg F G

ρ “ 'p q ' pρ q|G “ 'p q': Let be a ﬁnite group and let denote the regular -characterp1 of . Prove that: 'P ppGq 'P ppGq reg ÿ reg ÿ 1 Φ and Φ 1 IBr IBr Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Exercise 35.16 103

´ kG C “ px'; ψy 1 q Deduce from Theorem 35.11 that: p ';ψP ppGq |G| | p q ' P pGq 1 (a) thep inverse' of the Cartan matrixp of is IBr ; and

Exercise(b) 35.17Φ 1 for every IBr .

U kG P kG

(a) Let be a -module and let bepP;U a PIMq “ of . Prove' pg that´ q' pgq k kG |G| P U gPGp1 1 ÿ 1 dim Hom x ; yp1 F pGp1 q

(b) Prove that the binary operation is an inner product on Cl . Exercise 35.18 G “ A G p “ 5 Let : , the alternating group on 5A letters. Calculate thep Brauer character table, the Cartan

matrix and the decomposition matrix of for5 3. [Hints. (1.) Use the ordinary character table of and reduction modulo . (2.) A simple group does not have any Exerciseirreducible 35.19 Brauer character of degree 2.]

tr “ B g; h P Gp1 Deduce from Remark 35.12 that column orthogonality relations for the Brauer characters take the form Π Φ , i.e. given ´we have|CGpgq| g h G ; φpgq 'ph q “ : φP ppGq # ÿ 1 if and are -conjugate Φ IBr 0 otherwise Chapter 10. Blocks

blocks

We canG break down the representationp theorypF; ofO; finite kq groups into its smallest parts: the of the group algebra. First we will define blocks for arbitrary rings and then specify to the situation of a finite Notation:group and a splittingG -modular system p . pF; O; kq p F pGq pF; O; kq p Throughout,G denotes a finite group, a prime number. Wep let“ JpOq denoteP t aF;-modularO; ku systemG and we assume contains all exp -th roots of unity, so is a splitting -modular system for and all its subgroups (see Theorem 31.2). We write : . For Λ all References:Λ -modules considered are assumed to be free of finite rank over Λ.

Local representation theory

[Alp86] J. L. Alperin. The block theory of finite. group Vol. 11. algebras. Cambridge Vol. Studies I in Advanced Mathe- matics. Cambridge University Press, Cambridge, 1986. [Lin18] M. Linckelmann. Representations of groups . Vol. 91. London Mathemat- ical Society Student Texts. Cambridge University Press, Cambridge, 2018. [LP10] K. Lux and H. Pahlings. Representations of finite. Vol. groups 124. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010. [NT89] H. NagaoA and course Y. Tsushima. in finite group representation theory . Academic Press, Inc., Boston, MA, 1989. [Web16] P. Webb. . Vol. 161. Cambridge Studies in 36 TheAdvanced blocks Mathematics. of a ring Cambridge University Press, Cambridge, 2016.

A A Throughout this section we let denote an arbitrary associative ring with an identity element (de- notedProposition1 ). 36.1

(a) There is a bijection between A “ A ‘ ¨ ¨ ¨ ‘ Ar pr P Zě q (1) the set of decompositions A pA;Aq 1 1 A

of into a direct sum of -subbimodules (or equiv. of two-sided ideals of ), and

104 Skript zur Vorlesung: Modular Representation Theory WS 2020/21

105

A “ e ` ¨ ¨ ¨ ` er (2) the set of decompositions A 1 ZpAq 1 e “ A “ Ae ď i ď r of 1 into a sumi of orthogonalAi i idempotentsi of , ď i ď r A A pA;Aq in such a way that 1 and fori every 1 . ei (b) For each 1 , the direct summand of is indecomposable as an -bimodule if A pA;Aq and only if the corresponding central idempotent is primitive.

Proof :(c) The decomposition of into indecomposable -subbimodules is uniquely determined. A “ A ‘ ¨ ¨ ¨ ‘ Ar A pA;Aq ď i ď r ei P Ai 1 (a) Assume is a decompositionA “ e of` ¨into ¨ ¨ ` er : -bimodules. Then for each 1 , there exists such that a P A 1 1 a “ ¨ a “ pe ` ¨ ¨ ¨ ` e qa “ e a ` ¨ ¨ ¨ ` e a Furthermore, for any element A , r r

1 1 1 a “ a ¨ A “ ape ` ¨ ¨ ¨ ` erq “ ae ` ¨ ¨ ¨ ` aer : and ď j ď r ej a “ ae1j P Aj 1 e ; : : : ; er P ZpAq 1 r ai P Ai eiai “ ai ej ai “ j ‰ i ei “ Ai ei “ ei teiui“ 1 So for each 1 , we have ZpAq Ai “, andAei it followsď i ď inr particular that2 . r r 1 Also, if ttheneiui“ and 0 if . Hence, ZpAq1 and A “so i“ ei is a setAi of“ orthogonalAei p idempotentsA;Aq of , andA forď 1i ď r . A ‘ ¨ ¨ ¨ ‘ Ar ř Conversely, if A 1 ispA;A a setq of orthogonal idempotents of such that 1 1 , then 1 ñ: is anAi “ Ae-subbimodulei of for each 1 and f; h P ZispA aq direct sum decompositionei “ f ` h of intoAi “ Af-subbimodules.‘ Ah a P Af X Ah a “ af (b)" " Considera “ ah aand“ afh suppose“ a ¨ there“ exist orthogonalei idempotents Ai such that pA;A.q Then , where the sum is direct because if then ð and , henceAi “ Aei “ L ‘0L 0. Thus,L if isL not primitive, then pA;Aisq decomposable as an -bimodule.f P L zt u h P L zt u ei “ f ` h fh “ 1 2 1 2 " " Nowfh P L supposeX L “ that t u ei , whereAi and are two non-zero -submodules. 1 2 Then there exist 0 and 0 such that . Now, 0 since 1 2 f “ e f “ pf ` hqf “ f ` hf “ f ` “ f 0 . As is thei identity in , we get 2 2 2 h “ h f h e 0 i A “ A 2 ‘ ¨ ¨ ¨ ‘ Ar r P Zrě q A pA;Aq and similarly,L , hence and are orthogonalA idempotents, so is not primitive. A 1 1 (c) Supposex P thatL ( is a decomposition of into -subbimodules, and suppose that is an indecomposable direct summand of from a diﬀerent such decomposition of . x “ a ` ¨ ¨ ¨ ` a a P A ď i ď r Every can be written as r i i L Q e x “ a L “ p1L X A q ‘ ¨ ¨ ¨ ‘ pL X A q L L i i with r for each 1 , ď m ď r L “ L X Am Am 1 so . Hence Am andA this ispA;A a decompositionq of . Since is indecomposable there exists 1 such that . By the indecomposability of , this must be the whole of . Thus the decomposition of into -subbimodules is uniquely determined. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Deﬁnition 36.2 (Block, block idempotent, block algebra, belonging to a block) 106

pA;Aq Ai “ Aei ď i ď r blocks A ei (a) The uniquelyblock determined idempotents -bimodules (1 ) given by Theorem 36.1(c) are called the of and the correspondingblock algebras primitive centralA idempotents are called the associated . A M belongs to (or lies in) the block Ai “ Aei (The blocks are sometimes also called when is an algebra.) eiM “ M ej M “ j ‰ i (b) We say that an (indecomposable) -module if Exercise 36.3 and 0 for all .

A “ A ‘ ¨ ¨ ¨ ‘ Ar r P Zě A M A M M “ M ‘ ¨ ¨ ¨ ‘ Mr 1 1 Let ď i ď r ( Mi ) be the block decompositionAi A of and let be an arbitrary - 1 Amodule. Prove that admits a unique direct sumA decomposition where for each 1 the summand belongs to the block of . Deduce that every indecomposable Corollary-module 36.4 lies in a uniquely determined block of . A “ A ‘ ¨ ¨ ¨ ‘ Ar r P Zě A L M N A A ď i ď r 1 1 Let ( ) be the block decomposition of and let 0 0 be a s.e.s. of -modulesM and -module homomorphisms.Ai A Then,L forN each 1 Ai :

A lies inM the block of Aifi andA only and lie in .

In particular, if an -module lies in a block of , then so do all of its submodules and all of Proof : e P ZpAq A A its factori modules. i Ai “ Aei ei A Let be the primitive idempotent corresponding to . Now, by Deﬁnition 36.2 an -module belongs to the block if and only if external multiplication by is an -isomorphism on that module. Considering the commutative diagramL M N

– ei¨p´q ei¨p´q ei¨p´q – 0 0 L M N

M L N 0 0

it follows from the ﬁve-Lemma that this property holds for if and only if it holds for and .

37 p-Blocks of ﬁnite groups

WeExample now return 17 (Blocks to ﬁnite of groupsFG) and observe the following. FG FG

Since FGis semisimple, the block decomposition of is given by the Artin-Wedderburn Theorem. In particular, the blocks are matrix algebras and can be labelled by the isomorphism classes of simple -modules. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Remark 37.1 (Blocks of OG and kG) 107

OG rO{psG “ kG; x ÞÑ x “ x ` p ¨ OG The lifting of idempotents obtained in Theorem„ 32.3 tells us that the quotient morphism t ZpOGqu ÐÑ t ZpkGqu : induces a bijection e ÞÑ e primitive idempotents of primitive idempotents of OG “ e ` ¨ ¨ ¨ ` er OG kG “ e ` ¨ ¨ ¨ ` er 1 ThuskG a decomposition 1 of the identitykG element of into a sum of primitive 1 central idempotents correspondsO toG a decomposition 1kG of the identity element of into a sum of primitive central idempotents of . Therefore, by Proposition 36.1, there is a Warning bijection between the blocks of and the blocks of . blocks of finite groups algebras:idempotents The definitionsets of ofa block modules of a finite group may vary from one book (resp. article) to another and different authors use different definitions of which can, from text to text, be , , , sets of ordinary characters and/or Brauer characters. This may soundDefinition confusing 37.2 ( atBlocks first sight, of finite but groups, in general, principal the context block) makes it clear which kind of objects is meant. P tF; O; ku

blocks G p G; Gq Let Λ . Then:

(a) The of Λ are the uniquely determined indecomposable Λ Λ -bimodules defined p-block G OG by Theorem 36.1(c). kG OG kG (b) We define a of to be the specificationppGq of a block of p , understandingG alsoppkG theq corresponding block of kG, the corresponding idempotents of or of , or the modules which belongs to these blocks. We write Bl for the set of all -blocks of , resp. Bl Examplefor 18 the(p-block(s) set of all of blocks a p-group of ). G p G p kG kG If is a -group, then a unique -block.pkG; kGq Indeed, we have already observed that in this case, the trivial module is the unique simple module and hence a unique PIM, namely itself. So Theoremis certainly 37.3 also indecomposable as a -bimodule. K P tF; ku S fl T KG

S T KG Let and let be simple -modules. Then the following assertions are equivalent: KG S “ S ;:::;S “ T pm P Z q S S (a) and belong to the same block of . m ě i i` KG ď i ď m´ 1 2 1 (b) There exist simple -modules such that and are both Proof : composition factors of the same projective indecomposable -module forKG each 1 1.

ñ It is clear that all (a) and (b)m deﬁne equivalence relations on the set of simple -modules. We have toKG prove that it is the same relation.

(b)ñ(a): This followsS by induction on becauseB KG by Corollary 36.4B all“ theP ‘ composition ¨ ¨ ¨ ‘ Ps ‘ factorsQ ‘ of¨ ¨ ¨ a ‘ PIMQt of belong to the same block. Pi S 1 1 (a) (b): Assume belongsQj to the block of S. Decomposet “ inT PIMs where composition factors of the ’s are in relation (b) with but none of the composition factors of the ’s are in relation (b) with . To prove: 0. Clearly: then all simple modules Skript zur Vorlesung: Modular Representation Theory WS 2020/21

108 B S Pi Qj belongingPi to areBpP ini;Q relationj q “ (b)“ withBpQbyj ;P Corollaryiq 36.4 sinceď i ď itss projective coverď j mustď r be one of theP ‘’s. ¨ ¨ Now, ¨ ‘ Ps by construction,Q ‘ ¨ ¨ ¨ ‘ Q not composition factor of theB ’s can be a composition factor of the ’s, so Hom 0 Hom B for every 1 and every 1 KG . Thus 1 1 both t “ and are invariant under all -endomorphisms, in particular under left and right multiplication by elements of , so they must themselves be blocks of . The only possibility is 0. kG kG The eﬀect of this result is that the division of the simple -modules into blocksblock can be achieved in a purely combinatorial fashion, knowing the Cartan matrix of . The connection with a block matrix decomposition of the Cartan matrix is probably the origin of the use of the term in representation theory.Corollary 37.4 kG kG p On listing the simple -modulesp so that modules in eachp block occur together, the Cartan matrix of has a block diagonal form, with one block matrix for each -block of the group. Up to permutation of simple modules within -blocks and permutation of the -blocks, this is the unique decomposition Proofof : the Cartan matrix into block diagonal form with the maximum number of block matrices.

Given any matrix we may deﬁne an equivalence relation on the set of rows and columns of the matrix by requiring that a row be equivalentS to a column if and only if the entry in that row and columnPS is non-zero,S and extending this by transitivityPS to an equivalence relation. In the case of the Cartan matrix the row indexed by a simple module is in the same equivalence class as the column indexed by , because is a composition factor of . If we order the rows and columns of the Cartan matrix so that the rows and columns in each equivalence class come together, the matrix is in block diagonal form, with square blocks, and this is the unique such expression with the maximal number of blocks (up to permutation of the blocks and permutation of rows and columns within a block). It follows Theorem 37.3 that the matrix blocks biject with the blocks of the group algebra. 38 Defect Groups

kG OG From now on we will only discuss the blocks of p. (Analogous results hold for the corresponding blocks of .) To ﬁnish this chapter, for the sake of completeness, in this section and the next section weRemark give an 38.1 overview of important results concerning -blocks, mostly with sketches of proofs only. pkG; kGq B krG ˆ Gs G ˆ G Observe that any -bimodule becomes a left -module in a natural way via the -action pG ˆ Gq ˆ B Ñ B pg; hq; b ÞÑ gbh´

1 k ` krG ˆ ˘Gs conjugation action of G B kG krG ˆ Gs extendedkG by -bilinearity to the wholekr ofG ˆ Gs . This action is called the on . In particular, the group algebra itself is a left -module and the blocks of can be viewed as indecomposable -modules under this action. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Notation 38.2 109 G ÝÑ G ˆ G; g ÞÑ pg; gq G G ˆ G

TheoremWrite 38.3∆ : for the diagonal embedding of in . B P ppkGq B krG ˆ Gs pDq p D ď G D If G Bl , then every vertex of , considered as an indecomposable -module, has the form ∆ for some -subgroup . Moreover, is uniquely determined up to conjugation Proof : G ˆ G k G in . p q “ tpg ; g q P G ˆ G | g ¨ ¨ g´ “ u “ pGq : First, observe that GˆGpermutes the -basis of consisting of the group elements and 1 1 2 1 2 Stab 1 GˆG 1 1 ∆ kG “ k Ò pGq It follows that krGˆGs B pGq ∆ kG krG ˆ Gs Q B askrG a ˆ Gs -module, and hencepGq is relatively ∆ -projectiveQ p since it is by definition a directp summand ofD ď Gseen as Q “ p-module.Dq Therefore, by Definition 26.2, a vertex of (still considered as a -module) lies in ∆pDq. By Proposition 26.4, is a -group,G andˆ G thus thereD existsď aG -subgroup p such that ∆pD q, proving the firstB statement. pg ; g q P G ˆ G 1 Moreover, we know that ∆ is determined up to conjugacy in . Now, if is another 1 pD q “ pg ;g q pDq 1 2 -subgroup such that ∆ is a vertex of , then there exists such that 1 2 x P D pg ;g qpx; xq “ p g x; g x1q P pD q g x P D x P D g ∆ ∆ |D| “ |D | 1 2D “ D 1 2 1 1 1 and so for all , 1 ∆ . Hence for all . Finally, as 1 1 , it follows that . G

Note: As a defect group of a block is uniquely determined up to -conjugacy, it is clear that in fact allDeﬁnition defect groups 38.4 (Defect have the group, same defect order.)

B P ppkGq defect group B p D G pDq B Let Bl . krG ˆ Gs (a)A of is a -subgroup of such that ∆ is a vertex of considered as a B pd d P Z d defect B -module. ě 0 (b) If the defect groups of have order ( ) then is called the ofp .

Defect groups are useful and important because in some sense they measure how far a -block is from beingTheorem semisimple. 38.5 B kG D kG B D D If is a block of with defect group , then every indecomposable -module belonging to is Exerciserelatively 38.6 -projective, and hence has a vertex contained in .

B kG G D Prove Theorem 38.5. [Hint: Prove that , considered as -module via the conjugation action of , is relatively -projective.] Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Corollary 38.7 110 B kG B

Let be a block of with a trivial defect group. Then is a simple algebra, and in particular, is Proofsemisimple. : B D “ t u kG B t u B IfB has a trivial defect group 1 , then by TheoremB 38.5 every indecomposable -module belonging to is 1 -projective, i.e. projective. So is semisimple as all its modules are semisimple. But is an indecomposable algebra by deﬁnition. Hence is simple.

MainTheorem properties 38.8 (Green of the) defect groups are the following. We mention them without proofs. B kG D

P p G D g P C pDq Let be a block of with defect group . Then: G D “ P X gP (a) if is a Sylow -subgroup of containing , then there exists such that D p G ; D p N pDq Q “ O pN pQqq (b) contains every normal -subgroup of G ; p G

Example(c) 19is the largest normal -subgroup of , i.e. .

G p ppkGq “ tkGu D “ G Let be a -group. We already saw that in this case Bl . Then Theorem 38.8 shows that this block has a unique defect group . 39 Brauer’s 1st and 2nd Main Theorems

FinallyDeﬁnition we 39.1 present two main results due to Brauer. GˆG H ď G b P ppkHq B P ppkGq corresponds to b b | BÓHˆH and B kG B “ bG B Let , let bGBlis deﬁned. Then a block Bl if and only if , is the unique block of with this property. We then write . If such a block exists, G Propositionthen we say 39.2 that (Facts about b ). H ď G b kH D

bG D bG Let and let be a block of with defect group . H ď N ď G bN pbN qG bG bG “ pbN qG (a) If is deﬁned, then lies in a defect group of . C pDq ď H bG (b) If G , and , and are deﬁned, then .

Theorem 39.3 (Brauer’s First Main Theorem) 111

D ď G p H ď G NGpDq

„ Let tbe a -subgroupkH and let Dcontainingu ÝÑ t .kG Then there is a bijectionDu b ÞÑ bG Blocks of with defect group Blocks of with defect group bG Brauer correspondent b

ProofMoreover, (Sketch) : in this case is called the of .

Theorem 39.4 (ThisBrauer’s is a particular Second Maincase of Theorem the Green) correspondence. H ď G B kG b kH V B U b Q CGpQq ď H U G G G Let , let be aV blockÓH of b and let be a blockb “ of B . Suppose that is an indecomposable module in and is an indecomposable module in with vertex such that . If is Lemmaa direct 39.5 summand of , then is deﬁned and . S kG OppGq p G S kG krG{OppGqs kG Let be a simple -module. Then , the largest normal -subgroup of , acts trivially on . G Proof : OppGq G S Ó In particular, theE simple -modules are precisely the simpleOppGq -modules inﬂated to .

S G k k Since , we know from Cliﬀord’sÓOpp TheoremGq“ ‘ ¨that ¨ ¨ ‘ is semisimple, hence of the form

OppGq p P S since is a -group and hence has only one simple module up to isomorphism, namely the trivial Corollarymodule. 39.6 In other words, acts trivially on . The second claim follows immediately. B kG D kG B D Let be a block of with defect group . Then there exists an indecomposable -module Proofbelonging : N to“ NwithGpDq vertexb P. ppkNq p D B P ppkGq b D b D “ OppNq S WritekN : . Let Blb be a -block with defectD group andS let SBl be the Brauer correspondentkrN{Ds of . As is a defectPS group of , Sby Theorem k38.8rN{(c).Ds Now, let be N a simple -module belonging toN{D.p Then,PSq byPS LemmaN 39.5, acts triviallykN on . So can be viewedb as a simple -module and we let be the projective cover of seen as a -module. Then, Claim 1 D N pP q by Corollary 36.4, the inﬂationN{DInf S of to is an indecomposable -module belonging to . PS | krN{Ds : is a vertex of Inf . N pP q | N pkrN{Dsq “ kÒN Indeed: By deﬁnition , soN{D S N{D D

N N D Inf D ENInf kÒD ÓD N kD k D p and is therefore -projective. Now, since , it follows from Cliﬀord’s Theorem that is a direct N pP qÓN | k ‘ ¨ ¨ ¨ ‘ k : sum of -conjugates of the trivial -moduleN{D S, whichD are all again trivial since is a -group. So

N N Inf N{DpPSq ÓD D N N N D N{DpPSq N{DpPSq ÓD Therefore the indecomposable direct summandsN of Inf all have vertex . It follows then from N{DpPSq Lemma 27.1 that is a vertex of Inf , because Inf has at least one indecomposable direct summand with the same vertex as Inf . This proves Claim 1. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

N N 112 kG fp N{DpPSqq N{DpPSq kG D Next we observeN that the -Green correspondent Inf of Inf is certainly an indecom- Claim 2 fp N{DpPSqq B posable -module with vertex . : Inf belongs to . Indeed, this is clear by deﬁnition of the Green correspondent and Brauer’s First Main Theorem.

TheCorollary next result 39.7 shows that the converse of Corollary 38.7 also holds. B kG B

ProofA : blockB P ofppkGqis a simple algebra if and onlyB if has a trivial defect group. B kG B B If Bl has trivialB defect group, then is a simple algebra by CorollaryB 38.7. Suppose now that is a block of which is a simple algebra. Then is semisimple so all -modules are projective. Hence all indecomposable -modules have trivial vertices so, by Corollary 39.6, has a trivial defect Deﬁnitiongroup. 39.8 (Principal block)

principal block kG kG kG k B pkGq (a) The of is the block of to which the trivial -module belongs. This kG 0 p G full defect block is denote by .

Lemma(b) 39.9 A block of whose defect groups are the Sylow -subgroups of is said to have .

kG p G kG (a) Any indecomposable -module whose vertices are the Sylow -subgroups of belongs to B pkGq a block of of full defect. 0 Proof :(b) The principal block is a block of full defect. p G (a) is clear by Theorem 38.5. (b) is clear by (a) since the vertices of the trivial module are the Sylow -subgroups of . Appendix 1: Background Material Module Theory

This appendix provides you with a short recap of the notions of the theory of modules, which we will assume as known for this lecture. We quickly review elementary deﬁnitions and constructions such as quotients, direct sum, direct products, tensor products and exact sequences, where we emphasise the approachNotation via universal properties. R S unital associative : throughout this appendix we let and denote rings, and unless otherwise speciﬁed, all rings are assumed to be and . Commutative Algebra Most of the results are stated without proof, as they have been studied in the B.Sc. lecture References:. As further reference I recommend for example:

Advanced modern algebra. 2nd ed.

[Rot10] J. J. Rotman. Providence, RI: American Mathematical Society (AMS), 2010. A Modules, submodules, morphisms

Deﬁnition A.1 (Left R-module, right R-module, pR;Sq-bimodule)

left R-module pM; `; ¨q pM; `q ¨ R ˆ M ÝÑ M;pr; mq ÞÑ r ¨ m (a)A is anλ orderedR ÝÑ triple pMq , where is an abelian group and : is a binary operation such that the map r ÞÑ λprq “ λr M ÝÑ M; m ÞÑ r ¨ m : End ¨ scalar multiplication external composition law : : isright a ringR homomorphism.-module The operation is called a ¨ Morˆ anR ÝÑ M; pm; rq ÞÑ m ¨ r . (b)A pR;Sq-bimoduleis deﬁned analogouslypM; using`q a scalar multiplicationR : S on the right-hand side. (c) An ris¨ p anm ¨ abeliansq “ pr ¨ groupmq ¨ s @ r whichP R; @ iss P bothS; @ am leftP M:-module and a right -module, and which satisﬁes the axiom

1000 Skript zur Vorlesung: Modular Representation Theory WS 2020/21

1001 Convention R R R : Unless otherwise stated,rm in this lecturer ¨ m we always work with left modules. When no confusion is to be made, we will simply write " -module" to mean "left -module", denote -modules by their underlying sets and write instead of . Deﬁnitions for right modules and bimodules are similarDeﬁnition to those A.2 (R for-submodule left modules,) hence in the sequel we omit them. R-submodule R M U ď M r ¨ u P U @ r P R @ u P U

DeﬁnitionAn A.3 (Morphismsof an ) -module is a subgroup such that , . (homo)morphism R R-linear map R-homomorphism R ' M ÝÑ N A of -modules (or an , or an ) is a map of - ' modules : such that: 'pr ¨ mq “ r ¨ 'pmq @ r P R @ m P M (i) is a group homomorphism; and

(ii) , . ¨ R monomorphism Furthermore: epimorphism hook arrow ãÑ two-head Anarrow injective (resp. surjective) morphism of -modules is sometimes called a (resp. an ) and we often denote it with a " " (resp. a ¨ " "). R isomorphism R-isomorphism M – N R M N A bijective morphism of -modules is called an (or an ), and we ¨ write if thereR exists an -isomorphism betweenendomorphismand . automorphism A morphism from an -module to itself is called an and a bijective endomorphism is called an .

Notation RMod R R ModR R R RModS : We let denotepR;S theq category of left p-modulesR;Sq (with -linear maps as morphisms), we let denote the category of right -modules (with -linear maps as morphisms), and we let Exampledenote A.4 the category of -bimodules (with -linear maps as morphisms).

pM; `; ¨q

(a) Exercise: Check that Definition A.1(a) is equivalent to requiring that satisfies the following(M1) pM; axioms:`q (M2) pr ` r q ¨ m “ r ¨ m ` r ¨ m r ; r P R m P M is an abelian group; (M3) r ¨1 pm 2` m q “ 1r ¨ m ` 2r ¨ m 1r P2R m ; m P M for each and each ; (M4) prsq ¨ 1m “ r2¨ ps ¨ mq 1 2r; s P R m P M 1 2 for each and all ; (M5) R ¨ m “ m m P M for each and all . 1 for each . K K In other words, modules over rings satisfy the same axioms as vector spaces over fields. Hence: Vector spaces over a field are -modules, and conversely. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Z 1002

(b) Abelian groups are -modules, and conversely. R Exercise: check it! What is the external composition law?

(c) If the ring is commutative, then any right module can be made into a left module, and conversely. ' M ÝÑ N R p'q “ tm P M | 'pmq “ u Exercise: check it! Where does the commutativity come into play? N ' R M p'q “ 'pMq “ t'pmq | m P Mu ' (d) RIf : isN a morphism of -modules, then the kernel ker : 0 ´ ofM “isN an -submodule' of and the image Im : inverse mapof' is an -submoduleR of . 1 If and is invertible, then the inverse is the usual set-theoretic and is also an -homomorphism. Change of the base ring: ' S ÝÑ R R M Exercise: check it! S (e) if : is a ring homomorphism, then every -module can be endowed with the¨ structureS ˆ M ÝÑ of anM; ps;-module mq ÞÑ s with¨ m external“ 'psq ¨ compositionm : law given by

: :

Notation A.5Exercise: check it!

R M N R pM;Nq “ t' M ÝÑ N | ' R u

Given -modules and , we set Hom : : is an -homomorphism . ` pM;Nq ˆ pM;Nq ÝÑ pM;Nq This is an abelianR group for the pointwiseR addition of maps:R p'; ψq ÞÑ ' ` ψ M ÝÑ N; m ÞÑ 'pmq ` ψpmq : Hom Hom Hom N “ M pMq “ pM;Mq M R R : . R pMq M M In case , we write End : Hom for the set of endomorphisms of and Aut for the set of automorphisms of , i.e. the set of invertible endomorphisms of . Lemma-Deﬁnition A.6 (Quotients of modules) U R R M M{U R Let be an -submodule of an -module . The quotient group can be endowed with the structure of an -module in a naturalR wayˆ M via{U theÝÑ externalM{U composition law r; m ` U ÞÝÑ r ¨ m ` U

π M ÝÑ M{U;` m ÞÑ m˘ ` U R canonical natural homomorphism The canonical map : is -linear and we call it the (or Deﬁnition A.7) (Cokernel, coimage. ) ' P R pM;Nq cokernel ' R p'q “ N{ ' coimage ' R M{ ' Let Hom . The of is the quotient -module coker : Im , and the Theorem A.8of (Theis universal the quotient property-module of the quotientker . and the isomorphism theorems)

Universal property of the quotient ' M ÝÑ N R

(a) : Let : be a homomorphism of -modules. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

U R M U Ď p'q R 1003 ' M{U ÝÑ N ' ˝ π “ ' If is an -submodule of such that ker , then there exists a unique -module homomorphism : such that , or in other words such that the following ' diagram commutes: M N ö π D ' M{U !

'pm ` Uq “ 'pmq @ m ` U P M{U

1st isomorphism theorem U “ p'q Concretely, .

(b) : With the' notationM{ p' ofq (a), ÝÑ if p'q ker , then

R : ker Im 2nd isomorphism theorem U ;U R M U XU U `U is an isomorphism of -modules. R 1 2 1 2 1 2 (c) : If are -submodules of , then so are and , and there is an isomorphism ofpU-modules` U q{U – U {pU X U q :

3rd isomorphism theorem U Ď1 U 2 R2 1 1 2M R 1 2 (d) : If pM{U areq {pU-submodules{U q – M{U of: , then there is an isomorphism of -modules Correspondence theorem U R1 2 1 M 2

(e) tR : If X is anM | -submoduleU Ď Xu ÐÑ of , tR then there is a bijectionM{Uu X ÞÑ X{U ´ -submodulesπ pZofq Ð Z -submodules of [ 1 . B Free modules and projective modules

Free modules

Definition B.1 (Generating set / R-basis / finitely generated/free R-module) M R X Ď M M generated by X m P M R Let be an -module and let be a subset. Then: m “ xPX λx x λx P R (a) M “is x saidXyR to beM “ xPX Rx if every element may be written as an -linear ř combination , i.e. where is almost everywhere 0. In this case we write M finitelyř generated or . X R-basis basis X M M (b) is said to be if it admits a finite set of generators. in a unique way R xPX λx X λx P R (c) is an (or simply a ) if generates and if every element of can be written ř as an -linear combination (i.e. with almost everywhere 0). Skript zur Vorlesung: Modular Representation Theory WS 2020/21

M free R X |X| R-rank M 1004

Notation: M “ xPX Rx – xPX R (d) is called if it admits an -basis , and is called the of . À À Remark B.2 In this case we write .

Warning R R (a) : If the ring is not commutative, then it is not true in general that two diﬀerent X M X M S R bases of a free -module have the same number of elements.

(b) Let be a generating set for . Then, is a basis of if and only if is -linearly R R R R independent.

Proposition(c) If B.3is a (Universal ﬁeld, then property every -module of free modules is free.) ( -modules are -vector spaces in this case!) M R R X N R f X ÝÑ N R f M ÝÑ N Let be a free -module with -basis . If is an -module and : is a map (of sets), then there exists a unique -homomorphism p : such that the following diagram X f N commutes: ö D f incM !p f extending f by R-linearity

ProofWe : say thatRp is obtained by xPX λx x P M fp xPX λ. x xq “ xPX λx fpxq ř ř ř PropositionGiven B.4 an (Properties-linear combination of free modules) , set p : . The claim follows.

R M R

P R pP; ´q (a) Every -module is isomorphic toR a quotient of a free -module.

(b) If is a free -module, then Hom is an exact functor. Projective modules

Proposition-Deﬁnition B.5 (Projective module) P R

pP; ´q Let be an -module.R Then the following are equivalent: ψ P pM;Nq R (a) The functor RHom is exact. ψ˚ R pP;Mq ÝÑ R pP;Nq (b) If Hom is a surjective morphism of -modules, then the morphism of abelian π M ÝÑ P R π σ P ÝÑ M groups : Hom Hom is surjective. π ˝ σ “ P (c) If : is a surjective -linear map, then splits, i.e., there exists : such P R that Id .

(d) is isomorphic to a direct summand of a free -module. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

P P projective 1005

Example B.6 If satisﬁes these equivalent conditions, then is called . R “ Z Z Z

e R e “ e R – Re ‘ Rp ´ eq Re (a) If , then every submodule of a free -module is again free (main theorem on -modules). e ‰ ; 2 (b) Let be an idempotent in , that is . Then, 1 and is projective P P but not free if 0 1. iPI i i À (d) A direct sum of modules is projective if and only if each is projective. C Direct products and direct sums

tMiuiPI R iPI Mi tMiuiPI R ś Let be a family of -modules. Then the abelian group , that is the product of R ˆ Mi ÝÑ Mi seen as a family of abelian groups, becomes an -module via the following external composition law: iPI iPI ź ź r; pmiqiPI ÞÝÑ r ¨ mi iPI :

j P I πj ` iPI Mi ˘ÝÑ M`j ; pmiq˘iPI ÞÑ mj j Mj ś Furthermore, for each , we let : denotes the -th projection from theProposition product to C.1 the (Universal module property. of the direct product)

t'i L ÝÑ MiuiPI R R ' L ÝÑ iPI Mi πj ˝ ' “ 'j j P I If : is a family of -homomorphisms, then there exists a unique -homomorphism ś L : such that for every .

¨¨¨ 'k 'j ¨¨¨ ö ö iPI Mi

πkś πj { # Mk Mj

Thus, R L; Mi ÝÑ R pL;Miq iPI iPI ´ ź ¯ ź Hom f ÞÝÑ πi Hom˝ f iPI ` ˘

M M pm q m “ is an isomorphismiPI i of abelian groups.iPI i i iPI i mi “ i P I À ś directNow let sum be thet subgroupMiuiPI of consistingR of the elements such thatj P I 0 al- mostηj M everywherej ÝÑ iPI M (i.e.i; mj ÞÑ 0 exept for a ﬁnite subset of indicesMj ). This subgroup is called the of the family and is in fact an -submodule of the product. For each , we let À : denote the canonical injection of in the direct sum. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Proposition C.2 (Universal property of the direct sum) 1006

tfi Mi ÝÑ LuiPI R R ' iPI Mi ÝÑ L f ˝ ηj “ fj j P I If : is a family of -homomorphisms, then there exists a unique -homomorphism À L : such that for every C O [ .

¨¨¨ fk fj ¨¨¨ ö ö

iPI Mi ; c

ηk À ηj

Mk Mj

Thus, R Mi;L ÝÑ R pMi;Lq iPI iPI ´ à ¯ ź Hom f ÞÝÑ f ˝Homηi iPI ` ˘

Remarkis an C.3 isomorphism of abelian groups. |I| ă 8 iPI Mi “ iPI Mi À ś It is clear that if , then external. internal The direct sum as deﬁned above is often called an direct sum. This relates as follows with the usualDeﬁnition notion C.4 of (“Internal”direct direct sum: sums) M R N ;N R M M “ N ‘ N m P M m “ n ` n n P N n P N 1 2 1 2 Let be an -module and be two -submodules of . We write if every M “ N ‘ N 1 M2 “ N ` N1 1 N X 2N “2 t u can be written in a unique way as , where and .

1 2 1 2 1 2 InProposition fact C.5 (internal direct sum) if and only if and 0 . N ;N M M “ N ‘ N R

1 2 ' M ÝÑ N1 ˆ N2 “ N ‘ N If and are as above and then the homomorphism of -modules m “ n ` n ÞÑ pn ; n q 1 2 1 2 : (external direct sum) R 1 2 1 2 , M “ N is an isomorphism of -modules. iPI i À DThe above Exact generalises sequences to arbitrary internal direct sums .

plugging them Exact sequences constitute a very useful tool for the study of modules. Often we obtain valuable information about modules by in short exact sequences, where the other terms are known. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Deﬁnition D.1 (Exact sequence) 1007 ' ψ L ÝÑ M ÝÑ N R R exact at M ' “ ψ

Remark D.2 (Injectivity/surjectivity/short exact sequences) A sequence of -modules and -linear maps is called ( ) if Im ker .

' ' L ÝÑ M ðñ ÝÑ L ÝÑ M L

ψ ψ (a) M ÝÑ Nis injective ðñ0 M ÝÑ N ÝÑis exact at . N ' ψ (b) ÝÑ L ÝÑis surjectiveM ÝÑ N ÝÑ 0 isL exactM at N. ' ψ ψ R ψ M{ ' ÝÑ N; m ` ' ÞÑ ψpmq (c) 0 0shortis exact exact (i.e. sequence at , s.e.s.and ) if and only if is injective, is surjective and induces an -isomorphism : Im Im . ' P pL;Mq Such a sequenceR is called a ( for short). ' π (d) If Hom is an injectiveÝÑ morphism,L ÝÑ M ÝÑ then therep' isq ÝÑa s.e.s.

π 0 coker 0 ψ P pM;Nq where isR the canonical projection. i ψ (e) If Hom is a surjectiveÝÑ morphism,pψq ÝÑ thenM ÝÑ thereN isÝÑ a s.e.s.;

i 0 ker 0

Propositionwhere D.3 is the canonical injection. Q R

R pQ; ´q R Mod ÝÑ Ab left Let be an -module.' ψ Then the following holds: ÝÑ L ÝÑ M ÝÑ N ÝÑ R (a) Hom : is a exact covariant functor. In other words, if '˚ ψ˚ 0 / 0 isR p aQ;L s.e.sq of / -modules,R pQ;M thenq the/ inducedR pQ;N sequenceq

0 Hom Hom'˚ “ R pQ;'Homq '˚pαq “ ' ˝ α α P R pQ;Lq ψ˚ is an exact sequence of abelian groups. Here : Hom , that is for R p´;Qq R Mod ÝÑ Ab left every 'Hom ψ and similarly for . ÝÑ L ÝÑ M ÝÑ N ÝÑ R (b) Hom : is a exact contravariant functor. In other words, if ψ˚ '˚ 0 / 0 isR p aN;Q s.e.sq of -modules,/ R pM;Q thenq the induced/ R pL;Q sequenceq

˚ ˚ 0 Hom Hom' “ R p';QHomq ' pαq “ α ˝ ' ˚ α P R pM;Qq ψ is an exact sequence of abelian groups. Here : Hom , that is for every Hom and similarly for . R pQ; ´q R p´;Qq R pQ; ´q R p´;Qq right Exercise: verify that Hom and Hom are functors. Notice that Hom and Hom are not exact in general. Exercise: ﬁnd counter- examples! Skript zur Vorlesung: Modular Representation Theory WS 2020/21

1008 Lemma-Deﬁnition D.4 (Split short exact sequence) ' ψ ÝÑ L ÝÑ M ÝÑ N ÝÑ R split

A s.e.s. 0 0 of -modules is called if it satisﬁes one of the following ψ R D σ P pN;Mq ψ ˝ σ “ equivalent conditions: R N ' R D ρ P pM;Lq ρ ˝ ' “ (a) admits an -linear section, i.e. if Hom R such that Id ;L D R α M ÝÑ L ‘ N (b) admits an -linear retraction, i.e. if Hom such that Id ; ' ψ (c) an -isomorphism : / L such/ thatM the following/ N diagram/ commutes:

L ö α ö N 0 i p 0 Id/ L / L ‘ N / NId / ;

i p 0 0

Remark D.5where , resp. , are the canonical inclusion, resp. projection. σ M “ 'pLq ‘ σpNq ρ M “ 'pLq ‘ pρq If the sequence splits and is a section, then . If the sequence splits and is a Exampleretraction, D.6 then ker . Z

' π The s.e.s. of -modules / Z{ Z / Z{ Z ‘ Z{ Z / Z{ Z /

'pr sq “ pr0 s; r sq 2 π 2 2 2 0 '

deﬁned by 1 1 0 and where is the canonical projection onto the cokernel of is ' π split but the sequence / Z{ Z / Z{ Z / Z{ Z /

'pr sq “ pr sq 0 π 2 4 2 0 '

deﬁned by 1 2 and is the canonical projection onto the cokernel of is not split. Exercise: justify this fact using a straightforward argument. E Tensor products

Deﬁnition E.1 (Tensor product of R-modules) M R N R F Z M ˆ N G F Let be a right -module and let be a left -module. Let be the free abelian group (= free -module) with basis . Let be the subgroup of generated by all the elements pm ` m ; nq ´ pm ; nq ´ pm ; nq; @m ; m P M; @n P N; pm; n ` n q ´ pm; n q ´ pm; n q; @m P M; @n ; n P N; 1 2 1 2 1 2 pmr; nq ´ pm; rnq; @m P M; @n P N; @r P R: 1 2 1 2 1 2 and Skript zur Vorlesung: Modular Representation Theory WS 2020/21

1009 tensor product of M and N (balanced over R) M bR N “ F{G pm; nq P F M bR N m b n The , is the abelian group : . The Remark E.2 class of in is denoted by .

M bR N “ xm b n | m P M; n P NyZ M b N (a) R .

(b) In , wepm have` m theqb relationsn “ m b n ` m b n; @m ; m P M; @n P N; m b pn ` n q “ m b n ` m b n ; @m P M; @n ; n P N; 1 2 1 2 1 2 mr b n “ m b rn; @m P M; @n P N; @r P R: 1 2 1 2 1 2 and m b “ “ b n @ m P M @ n P N p´mq b n “ ´pm b nq “ m b p´nq @ m P M @ n P N In particular, 0 0 0 , and Deﬁnition E.3 (R-balanced, . map) M N A f M ˆ N ÝÑ A R-balanced Let and be as above and let be an abelian group. A map : is called if fpm ` m ; nq “ fpm ; nq ` fpm ; nq; @m ; m P M; @n P N; fpm; n ` n q “ fpm; n q ` fpm; n q; @m P M; @n ; n P N; 1 2 1 2 1 2 fpmr; nq “ fpm; rnq; @m P M; @n P N; @r P R: 1 2 1 2 1 2

Remark E.4

t M ˆ N ÝÑ M bR N; pm; nq ÞÑ m b n R Proposition E.5 (Universal property of the tensor product) The canonical map : is -balanced. M R N R A R f M ˆ N ÝÑ A Z f M bR N ÝÑ A Let be a right -module and let be a left -module.f For every abelian group and every M ˆ N /; A -balanced map : there exists a uniqueö -linear map : such that t the following diagram commutes: f M bR N

Proof : { M ˆ N ÝÑ F π F ÝÑ F{G Z Z f F ÝÑ A Let :f ˝ { “ f fdenoteR the canonical inclusion,G andĎ letpfq : denote the canonical projection. By the universal property of the free -module, there exists a unique f -linearF{G ÝÑ map A˜ : suchf ˝ π that“ f ˜ . Since is -balanced, we have that ker ˜ . Therefore, the universal property of the quotient yields the existence of a unique homomorphismf of abelian groups : such that M ˆ N /: A ˜ J : { f t F ˜

π f M bR N – F{G t “ π ˝ { f ˝ t “ f ˝ π ˝ { “ f ˝ { “ f

Clearly , and hence ˜ . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Remark E.6 1010 M N

tMiuiPI R M R N R Let and be as in Deﬁnition E.1. tNj uiPJ R (a) Let be a collection of right -modules, be a right -module, be a left -module M b N – pM b Nq and be a collection of left -modules.i R Then, wei R have iPI iPI à à M bR Nj – pM bR Nj q: jPJ jPJ à à

(This is easily proved using both the universal property of the direct sum and of the tensor R b N – N r b n ÞÑ rn product.) R M bR R – M m b r ÞÑ mr (b) There are natural isomorphisms of abelian groups given by via , and P R R X N b P – N via . R xPX P R R X P bR M – M xPX À (c) It follows from (b), that if is a free left -module with -basis , then , Q À and if is a free right -module with -basis , then . M pQ;Rq N R M b N (d) Let be a third ring. Then we obtain module structures on the tensorR product as follows: Q (i) If is a -bimodule and a left -module, then can be endowed with q ¨ pm b nq “ q ¨ m b n @q P Q; @m P M; @n P N: the structure of a left -module via

M R N pR;Sq M bR N S (ii) If is a right -module and an -bimodule, then can be endowed with pm b nq ¨ s “ qm b n ¨ s @s P S; @m P M; @n P N: the structure of a right -module via

M pQ;Rq N pR;Sq M bR N pQ;Sq (iii) If is a -bimodule and an -bimodule. Then can be endowed with the structure of a -bimodule via the external composition laws deﬁned in (i) R R pR;Rq and (ii). M bR N R (e) Assume is commutative. Then any -module can be viewed as an -bimodule. Then, K M N K in particular, becomes an -module (both on the left and on the right). K txiuiPI tyj ujPJ M bK N K K (f) Fortxi b instance,yj upi;jqPIˆ itJ follows from (e) that if is a ﬁeld and and are -vector spaces with -bases and resp., then is a -vector space with a -basis given by Tensor product of morphisms: f M ÝÑ M1 R . g N ÝÑ N1 R 1 1 (g) Let Z: fbeb g a morphismM bR N ÝÑ of rightM bR-modulesN and pf b: gqpm b nq “befp am morphismq b gpnq of left -modules. Then, by the universal property of the tensor product, there exists a unique -linear map : such that . Skript zur Vorlesung: Modular Representation Theory WS 2020/21

Exercise E.7 1011

R I R M R R R{I bR M – M{IM (a) Assume is a commutative ring and is an ideal of . Let be a left -module. Prove that m; n Z{nZ b Z{mZ Q b Q Q{Z b Q there is an isomorphism of left -modules Z . Z Z K U;V K (b) Let be coprime positive integers. Compute , , and . K

(c) Let be a ﬁeld and let be ﬁnite-dimensional˚ -vector spaces. Prove that there is a natural isomorphism of -vector spaces:K pU;V q – U bK V:

Hom Proposition E.8 (Right exactness of the tensor product)

N R ´ bR N ModR ÝÑ Ab M R M b ´ Mod ÝÑ Ab (a) Let be a left -module. Then R : R is a right exact covariant functor.

Remark(b) E.9Let be a right -module. Then : is a right exact covariant functor.

´ bR N M bR ´

The functors and are not left exact in general. F Algebras

algebras

DeﬁnitionIn this lecture F.1 we (Algebra aim at) studying modules over speciﬁc rings, which are in particular . R

R-algebra pA; `; ¨; ˚q Let be a commutative ring. (A1) pA; `; ¨q (a) An is an ordered quadruple such that the following axioms hold: (A2) pA; `; ˚q R is a ring; (A3) r ˚ pa ¨ bq “ pr ˚ aq ¨ b “ a ¨ pr ˚ bq @ a; b P A @ r P R is a left -module; and f A Ñ B R algebra homomorphism , . f R (b) A map : between two -algebras is called an iﬀ: f (i) is a homomorphism of -modules; Example F.2(ii) (Algebrasis a ring) homomorphism. R R ¨ ˚ (a) The ring itself is an -algebra. n P Zě MnpRq n ˆ n R R [The internalR composition law " " and the external composition law " " coincide in this case.] 1 (b) For each the set of -matrices with coeﬃcients in is an -algebra for its usual -module and ring structures. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

R 1012

K n P Z K rX ;:::;X s K [Note: in particular -algebras need notě be commutative rings in general!]n K 1 1 (c) Let be a ﬁeld. Then for each the polynom ring is a -algebra for R C Q C R its usual -vector space and ring structure. Z (d) and are -algebras, is an -algebra, . . .

(e) Rings are -algebras. ExampleExercise: F.3 (Modules Check over it! algebras)

n A “ MnpRq ñ R A A ˆ Rn ÝÑ Rn; pB; xq ÞÑ Bx (a) is an -module for the external composition law given by left matrix K V K V A A “ pV q multiplication . K

(b) If is a ﬁeld and a -vector space, then becomes an -algebra for : End together with the external compositionA ˆ V ÝÑ law V; p'; vq ÞÑ 'pvq :

A M R Exercise: Check it! R ÝÑ A; r ÞÑ r ˚ A (c) An arbitrary -module can be seen as an -module via a change of the base ring since Exercise F.4 1 is a homomorphism of rings by the algebra axioms. R

M;N R Let be a commutative ring. pMq (a) Let Rbe -modules. Prove that: R

(1) End , endowed withR thepM;N pointwiseq additionR of maps and the usual composition of maps, is a ring. (Note that the commutativity of is not necessary!) (2) The abelian group Hom is a left -module for the external composition law deﬁned byprfqpmq “ fprmq “ rfpmq @ r P R; @f P R pM;Nq; @m P M:

A R : M A Hom R pMq ApMq R (b) Let now be an -algebra and be an -module. Prove that End and End are -algebras. Appendix 2: The Language of Category Theory

This appendix gives a short introduction to some of the basic notions of category theory used in this lecture. References:

Categories for the working mathematician

[Mac98] S. Mac Lane.An introduction to homological algebra . Second. Vol. 5. Springer-Verlag, New York, 1998. [Wei94] C. A. Weibel. . Vol. 38. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1994. G Categories

Deﬁnition G.1 (Category) category C ‚ C objects A consists of: ‚ CpA;Bq morphisms pA;Bq a class Ob of , ‚ composition function a set Hom of for every ordered pair of objects, and CpA;Bq ˆ CpB;Cq ÝÑ CpA;Cq a pf; gq ÞÑ g ˝ f pHomA;B;Cq Hom Hom

for each ordered triple of objects, (C1) Unit axiom A P C identity morphism A P CpA;Aq satisfying the following axioms:f P CpA;Bq B P C

: for each object Obf ˝, thereA “ f exists“ B ˝ anf : 1 Hom such that for every Hom for all Ob , (C2) Associativity axiom f P pA;Bq g P pB;Cq h P pC;Dq 1C 1 C C A;B;C;D P C : for every hHom˝ pg ˝ fq “ p, h ˝ gqHom ˝ f : and Hom with Ob ,

1013 Skript zur Vorlesung: Modular Representation Theory WS 2020/21

1014

Remark G.2 Let us start with some remarks and examples to enlighten this deﬁnition: C

pA;Bq (a) Ob need not be a set! C f (b) The only requirement onf HomA ÝÑ B isA ÝÑ thatB it be a set, andf P it is allowedCpA;Bq to be empty. arrows morphisms A P C A P C (c) It is common to write : or instead of Hom , and to talk about P pA;Aq f P pA;Aq instead of A . It isC also common to write " " instead of " A Ob ". C fA “ fA ˝ A “ A (d) The identity morphism 1 Hom is uniquely determined: indeed, if Hom Examplewere G.3 a second identity morphisms, then we would have 1 1 .

C “ 1

(a) : category with one object and one morphism‚ (the identity morphism): ‚ 1

C “ 2

(b) : category with two objects and three morphism, where two of them are identity morphisms and the third one goes from one object to the other: A B A B 1 1

G CpGq CpGq “ t‚u CpGqp‚; ‚q “ G

(c) A group can be seen as a category with one object: Ob , Hom n ˆ m k n; m (notice that this is a set) and composition is given by multiplication in the group. Matk Matk “ Zą n ÝÑ m n m m ˆ n (d) The -matrices with entries in a ﬁeld for ranging over the positive integers form 0 a category : Ob , morphisms from to are the -matrices, Exampleand G.4 compositions(Categories and are algebraicgiven by the structures ordinary) matrix multiplication.

C “ Set category of sets

(a) , the : objects are sets, morphisms are maps of sets, and composition C “ Veck category of vector spaces over the ﬁeld k k is the usual composition of functions. k (b) , the : objects are -vector spaces, mor- C “ Top category of topological spaces phisms are -linear maps, and composition is the usual composition of functions. (c) , the : objects are topological spaces, morphisms are C “ Grp category of groups continous maps, and composition is the usual composition of functions. (d) , the : objects are groups, morphisms are homomorphisms of groups, and composition is the usual composition of functions. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

C “ Ab category of abelian groups 1015

(e) , the : objects are abelian groups, morphisms are homomor- C “ Rng category of rings phisms of groups, and composition is the usual composition of functions.

(f) , the : objects are rings, morphisms are homomorphisms of rings, C “ Mod category of left R-modules left R andR composition is the usual composition of functions. R (g) , the : objects are modules over the ring , morphisms C “ Mod category of left R-modules right R are -homomorphisms,R and composition is the usual composition of functions. R (g’) , the : objects are modules over the ring , C “ Mod category of pR;Sq-bimodules pR;Sq morphismsR S are -homomorphisms, and composition is the usual composition of functions. R S pR;Sq (g”) , the : objects are -bimodules over the rings and , morphisms are -homomorphisms, and composition is the usual composition of ::: functions.

Deﬁnition(h) Examples G.5 (Monomorphism/epimorphism of your own )

C f P CpA;Bq f monomorphism g ; g C ÝÑ A Let be a category and let Hom be a morphism. Then is called 1 2 (a)a iﬀ for all morphismsf ˝ g “ f ˝ g: ùñ g ,“ g :

epimorphism 1g ; g B2 ÝÑ C1 2

1 2 (b) an iﬀ for all morphismsg ˝ f “ g ˝: f ùñ g , “ g :

1 2 1 2 Remark G.6

In categories, where morphismsRMod Mod areR set-theoreticRModS maps, then injective morphisms are monomorphisms, andWarning surjective morphisms are epimorphisms. In module categories ( , , , . . . ), the converse holds as well, but: : It is notRng true in general, that all monomorphisms{ Z ÝÑ Q must be injective, and all epimorphismsC mustg be; g surjective.P RngpQ;Cq g For example in , the canonical injection { : / is an epimorphism. Indeed, if is a ring 1 2 Z / Q / C and Hom g2 g ˝ { “ g ˝ { g “1 g { 1 2 1 2 are such that , then we must have by the universal property of the ﬁeld of fractions. However, is clearly not surjective. Skript zur Vorlesung: Modular Representation Theory WS 2020/21

H Functors 1016 Deﬁnition H.1 (Covariant functor) C D covariant functor F C ÝÑ D

‚ F C ÝÑ D ;X ÞÑ FpXq Let and be categories. A : is a collection of maps: ‚ F pA;Bq ÞÑ pFpAq;FpBqq A;B: Ob C Ob D , and

: Hom Hom , f g satisfying:A ÝÑ B ÝÑ C C Fpg ˝ fq “ Fpgq ˝ Fpfq Fp q “ A P C (a) If A FpAq are morphisms in , then ; and

Deﬁnition(b) H.21 (Contravariant1 for every functor)Ob . C D contravariant functor F C ÝÑ D

‚ F C ÝÑ D ;X ÞÑ FpXq Let and be categories. A : is a collection of maps: ‚ F pA;Bq ÞÑ pFpBq;FpAqq A;B: Ob C Ob D , and

: Hom Hom , f g satisfying:A ÝÑ B ÝÑ C C Fpg ˝ fq “ Fpfq ˝ Fpgq Fp q “ A P C (a) If A FpAq are morphisms in , then ; and

Remark(b) H.31 1 for every Ob .

Often in the literature functors are deﬁned only on objects of categories. When no confusion is to Examplebe made H.4 and the action of functors on the morphism sets are implicitely obvious, we will also adopt this convention. Q P pRModq pQ; ´q Mod ÝÑ Ab Let Ob . Then R R M ÞÑ R pQ;Mq Hom : Hom , R p´;Qq RMod ÝÑ Ab is a covariant functor, and M ÞÑ R pM;Qq Hom : Hom , is a contravariant functor. Exact Functors.

We are now interested in the relations between functors and exact sequences in categories where it makes sense to deﬁne exact sequences, that is categories that behave essentially like module categories Skript zur Vorlesung: Modular Representation Theory WS 2020/21

1017 RMod abelian categories such as . These are the so-called . It is not the aim, to go into these details, ‚ Mod but roughly speaking abelian categoriesR are categories satisfying the following properties: ‚ Mod they have a zero object (in : the zeroR module) ‚ Mod R they have products and coproductsR (in : products and direct sums) ‚ Mod they have kernels and cokernels (in : the usual kernels and cokernelsR of -linear maps)

Deﬁnitionmonomorphisms H.5 (Pre-additive are kernels categories/additive and epimorphisms functors are cokernels) (in : satisﬁed)

C pre-additive

F C ÝÑ D additive F (a) A category in which all sets of morphisms are abelian groups is called . A;B A;B P C (b) A functor : between pre-additive categories is called iﬀ the maps Deﬁnitionare H.6 homomorphisms (Left exact/right of exact/exactgroups for all functors)Ob . F C ÝÑ D g ÝÑ A ÝÑf B ÝÑ C ÝÑ C F Let : be a covariant (resp. contravariant) additive functor between two abelian categories, Fpfq Fpgq Fpgq Fpfq and letleft0 exact ÝÑ FpAq ÝÑ0FbepBq aÝÑ s.e.s.Fp ofC objectsq andÝÑ morphismsFpCq ÝÑ in F.pB Thenq ÝÑ FispA called:qq

(a) if 0 Fpfq Fpgq (resp. 0 Fpgq Fpfq ) is an exactright exact sequence.FpAq ÝÑ FpBq ÝÑ FpCq ÝÑ FpCq ÝÑ FpBq ÝÑ FpAqq ÝÑ

(b) if Fpfq Fpgq 0 (resp. Fpgq Fpfq 0) is an exact sequence.ÝÑ FpAq ÝÑ FpBq ÝÑ FpCq ÝÑ ÝÑ FpCq ÝÑ FpBq ÝÑFpAqq ÝÑ

R pQ; ´q R p´;Qq R pQ; ´q Q R p´;Qq Q The functors Hom and Hom of Example H.4 are both left exact functors. Moreover Hom is exact if and only if is projective, and Hom is exact if and only if is injective. Index of Notation

General symbols C Fq q M field of complex numbersM pfq finite field with felements Id p'q identity map on the set' ImN image of the map kerN kernel of the morphism O the natural numbers without 0 0 P the natural numbers withZ 0 Q discrete valuation ring Qp the primep numbers in R field of rational numbers Z field of -adic numbers Zěa; Ząa; Zďa; Zăa tfieldm P ofZ | realm ě numbersa m ą a; m ě a; m ă aqu Zp ring of integerp numbers |X| (resp.X δij ring of -adic integers cardinality of the set Kronecker’s delta Ť union š disjoint union Ş ˆ intersection ř ¸ summation symbol ś ‘ , cartesian/direct product b semi-direct product H direct sum @ tensor product D empty set – for all a | b a - bthere a existsb a b pa; bq isomorphisma b pF; O,; kq p divides , does not divide f|S gcd of and f S ãÑ -modular system restriction of the map to the subset injective map surjective map Group theory pGq G An n AutCm automorphism group ofm the group CGpxq alternating group on lettersx G CGpHq cyclic group of order in multiplicativeH G notation D n centraliser of the elementn in δ G Ñ G ˆ G centraliser of the subgroup in 2 pAq dihedral group of order 2 A G{:N diagonal map G N EndnpK q endomorphism ring of theK abelian group HgL quotientpH;Lq group modulo rGLHzG{Ls generalpH;L linearq group over H ď G H ă GH -double cosetG N Ĳ GNset of -double cosetG representatives NGpHq, is a subgroupH ofG , resp. a proper subgroup N ¸θ H is a normal subgroupN H θ Sn normaliser of in n npK q semi-direct product of inK w.r.t. Z{mZ symmetric group on mletters x ´ SLg special linearg groupx overgxg xgy Ď G cyclic groupG of order in additiveg1 notation |G H| conjugate of by , i.e.H G rG{Hs subgroup of generated by H x P:G{N index ofx theP G subgroup in G{N t u set of left coset representatives of class of in the quotient group Module1 , 1 theory trivial group R pM;Nq R M N R pMq R R M HompMq -homomorphisms fromM to EndKG -endomorphism ring of theG -module K hdε KG ÝÑ K head of the module IpKGq group algebra of the group over the commutative ring JpR: q augmentation map R M | NMaugmentation ideal N M bR N Jacobson radical ofM the ringN R G M G is a direct summand of M MG Gtensor product of and balancedM over G G M ÓH H pMq -fixed pointsM of theG moduleH H G M ÒG H pMq -cofixed pointsM of theH moduleG G G{N, pResMq restriction ofM fromG{Nto G Rˆ , Ind induction of fromR to InfR˝ inflation of R from to R pMq units of the ring M pMq regular left -moduleM on the ring rad radical of the module soc socle of the module xXyR R X F V F bO V -module generatedR by the set extension of scalars Z(R)Character and centre Block of Theory the ring bG b C G F pGq F pGp1 q Brauer correspondent ofG Gp1 ppGq Cartan matrix of ClGp1 , Cl thep class functions on G or DecF pGq decomposition matrixF G ppGq -regular elementsp of G Irrχ ordinary irreducible -characters of IBrρ irreducible -Brauer characters of reg ' regular character ' P pGq reg regular representation CategoryΦ Theory projective indecomposable character associated to IBr C C CpA;Bq A B ObSet objects of the category HomVeck morphisms from to k Top the category of sets Grp the category of vector spaces over the field Ab the category of topological spaces Rng the category of groups RMod the category of abelianR groups ModR the category of ringsR RModS the category ofp leftR;Sq-modules the category of left -modules the category of -bimodules