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Topics in Group Representation Theory 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.
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