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