Topics in Group Representation Theory Peter Webb October 2, 2017 Contents 0.1 Representations of Sr ............................1 0.2 Polynomial representations of GLn(k)...................1 0.3 Representations of GL(Fpn ).........................2 0.4 Functorial methods . .2 1 Dual spaces and bilinear forms 3 2 Representations of Sr 4 2.1 Tableaux and tabloids . .4 2.2 Permutation modules . .5 2.2.1 Exercise . .7 2.3 p-regular partitions . .7 2.4 Young modules . .9 3 The Schur algebra 11 3.1 Tensor space . 11 3.2 Homomorphisms between permutation modules . 14 3.3 The structure of endomorphism rings . 15 3.4 The radical . 17 3.5 Projective covers, Nakayama's lemma and lifting of idempotents . 20 3.6 Projective modules for finite dimensional algebras . 24 3.7 Cartan invariants . 26 3.8 Projective and simple modules for SF (n; r)................ 28 3.9 Duality and the Schur algebra . 30 4 Polynomial representations of GLn(F ) 33 4.0.1 Multi-indices . 39 4.1 Weights and Characters . 40 5 Connections between the Schur algebra and the symmetric group: the Schur functor 47 5.1 The general theory of the functor f : B -mod ! eBe -mod . 51 5.2 Applications . 51 i CONTENTS ii 6 Representations of the category of vector spaces 54 6.1 Simple representations of the matrix monoid . 55 6.2 Simple representations of categories and the category algebra . 57 6.3 Parametrization of simple and projective representations . 60 6.4 Projective functors . 63 7 Bibliography 68 Landmark Theorems 0.1 Representations of Sr λ Theorem 0.1.1. The simple representations D of Sr over Fp are parametrized by p-regular partitions λ of r. They are self-dual and absolutely irreducible. Other properties: The Specht module Sλ has one composition factor Dλ and all µ others are D with µ D λ. Theorem 0.1.2. As µ ranges over partitions of r, the indecomposable summands Y λ of M µ = "Sr are parametrized by the partitions λ of r. Each permutation module Fp Sµ µ µ λ M has a summand Y with multiplicity 1, and all other summands M have λ D µ. Other properties: The M λ have a Specht filtration. M λ is projective if and only if λ0 is p-regular. We will avoid the usual things done in characteristic zero in other courses: con- struction of the character table, hook length formula, Murnaghan-Nakayama etc etc. 0.2 Polynomial representations of GLn(k) n ⊗r Let k be a field of characteristic p and E = k . Sr acts on E by permuting the positions of the tensor factors, and E⊗r is a direct sum of permutation modules M λ. We ⊗r define Sk(n; r) = EndkSr (E ). This is the Schur algebra associated to this situation. Theorem 0.2.1. Sk(n; r) is quasi-hereditary. When n ≥ r all the Sk(n; r) are Morita equivalent. The simple Sk(n; r)-modules are parametrized by partitions of r into at most n parts. Theorem 0.2.2. The polynomial representations of GL(n(k)) are direct sums of ho- mogeneous polynomial representations of various degrees r. Theorem 0.2.3. When k is infinite the algebra homomorphism k[GLn(k)] ! Sk(n; r) is surjective. The polynomial representations of degree r are the same as the represen- tations of Sk(n; r). When n ≥ r the simple polynomial representations of degree r are parametrized by the partitions of r. 1 CONTENTS 2 0.3 Representations of GL(Fpn ) Theorem 0.3.1. The simple Fpn GL(Fpn )-modules are parametrized by `weights' and are absolutely simple. Thus, for any field k of characteristic p, the simple kGL(Fpn )- n modules can all be written in Fp . 0.4 Functorial methods Writing Veck for the category of finite dimensional vector spaces over k we consider the category Funk of functors Veck ! Veck. This is an abelian category. For example, 17 6 the functor V 7! S V and the functor V 7! Λ V lie in Funk. Theorem 0.4.1. The simple objects in Funk are parametrized by pairs (n; W ) where n ≥ 0 and W is a simple representation of GLn(k). The corresponding simple functor sends kn to W and is zero on spaces of dimension < n. Each value on km is a simple module for GLm(k) or zero. Chapter 1 Dual spaces and bilinear forms This follows the first section of James' book and is quite incomplete. ∗ Let M be a finite dimensional vector space over a field k and put M = Homk(M; k). Let V be a subspace of M. We may take a basis e1; : : : ; ek for V and extend it to a ∗ basis e1; : : : ; em for M, so that M has a dual basis 1; : : : ; m with i(ej) = δij. Thus dim M = dim M ∗. ◦ ∗ We put V = ff 2 M fjV = 0g. ◦ ◦ Lemma 1.0.1. V has basis k+1; : : : ; m, so that dim V + dim V = m. Proposition 1.0.2. The vector space of bilinear forms h−; −i : M × M ! k is iso- morphic to Hom(M; M ∗). Proof. Suppose that h−; −i : M × M ! k is a bilinear form on M. There is a map Lemma 1.0.3. A bilinear form is non-singular if and only if the corresponding map θ is injective. Thus if dim M is finite, it is equivalent to require that θ : M ! M ∗ be an isomorphism. We define V ? = fx 2 M hx; vi = 0 for all v 2 V g. Lemma 1.0.4. If U ⊆ V , W are subspaces of M then V ? ⊆ U ? and (U + W )? = U ? \ V ?. Proposition 1.0.5. If h−; −i is non-singular and dim M is finite then θ : M ! M ∗ maps V ? isomorphically to V ◦ and hence dim V + dim V ? = dim M. We have V = V ?? and (U \ V )? = U ? + V ?. 3 Chapter 2 Representations of Sr The approach and results are taken directly from James' book. 2.1 Tableaux and tabloids A partition of r is a list of positive integers λ = [λ1; λ2; λ3;:::] with λ1 ≥ λ2 ≥ λ3 ≥ ::: P and λi = r. Partitions are partially ordered by the dominance relation: we say that Pj Pj λ = [λ1; λ2; λ3;:::] dominates µ = [µ1; µ2; µ3;:::] if for all j, i=1 λi ≥ i=1 µi. For each partition we have a Young subgroup Sλ = Sλ1 × Sλ2 × · · · . The set of right cosets SλnSr is a transitive Sr-set whose elements can be described in a certain way, by means of tabloids. We first define a λ-tableau to be a Young diagram of shape λ filled with the numbers f1; : : : ; rg, such as 5 1 2 t = 4 3 for the partition λ = [3; 2]. The symmetric group Sr permutes the λ-tableaux by acting on their entries. Let Rt and Ct denote the row and column stabilizers of t, so that in this example Rt = Sf5;1;2g × Sf4;3g;Ct = Sf5;4g × Sf1;3g × Sf2g: −1 −1 Lemma 2.1.1. If t is a λ-tableau and π 2 Sr then Rtπ = π Rtπ and Ctπ = π Ctπ. We put an equivalence relation on λ-tableaux by saying that two are equivalent if the numbers in each row are the same. Thus the equivalence class of t is tRt. We write the equivalence class as ftg and denote it pictorially by leaving out the vertical lines: 5 1 2 ftg = 4 3 Such an equivalence class is called a λ-tabloid. The one above is the same as the tabloid 1 2 5 3 4 4 CHAPTER 2. REPRESENTATIONS OF SR 5 and evidently we can write tabloids with the entries increasing along each row. Now Sr permutes the set of tabloids by acting on the entries. The stabilizer of the λ-tabloid with the numbers f1; : : : ; rg written along the rows in order is the Young subgroup Sλ and the action on the λ-tabloids is transitive. Hence: Lemma 2.1.2. The set of λ-tabloids is isomorphic to SλnSr as a Sr-set. The number of λ-tabloids is r! λ1!λ2! ··· The Sr-sets that arise this way include examples such as the set of unordered tuples of elements of f1; : : : ; rg of some given length, or the set of ordered tuples of some given length, or combinations of these possibilities. 2.2 Permutation modules Over a ring k define M λ to be the permutation module on the set of λ-tabloids. Thus M λ =∼ M "Sr , and M λ is generated as a kS -module by any single tabloid. If t is a Sλ r λ-tableau let κt be the element of the group algebra kSr that is the signed sum of the elements of C . Thus t X κt = (sgn π)π: π2Ct λ We define et = ftgκt as an element of M . In the case of the example 5 1 2 t = 4 3 we have 5 1 2 4 1 2 5 3 2 4 3 2 et = − − + : 4 3 5 3 4 1 5 1 The Specht module Sλ for the partition λ is defined to be the submodule of M λ spanned by the polytabloids. Proposition 2.2.1. Let t be a λ-tableau and π 2 Sr. 1. et is a linear combination of tabloids with ±1 coefficients. −1 2. κtπ = π κtπ and etπ = etπ. λ 3. S is a kSr-module and is generated as a kSr-module by any single polytabloid.