An Introduction to the Representation Theory of Finite Groups by G. Hiss, R. Kessar and B. Külshammer Last Updated: 30/05/2012 Introduction These notes are adapted from three short courses given by the above at a summer school entitled “An Introduction to the Representation Theory of Finite Groups" held at RWTH Aachen University between the 27th and 29th of September 2010. The three short courses were entitled • “An Introduction to Ordinary Representation Theory” given by Hiss, • “An Introduction to Modular Representation Theory” given by Külshammer, • “An Introduction to Block Theory” given by Kessar. The summer school was organised by Jürgen Müller, Natalie Naehrig and Gabriele Nebe as part of the DFG priority program in representation theory. These notes follow very closely the original lectures given by the above at this summer school, although some changes have been made. Mainly these changes are in the addition of some background material but also some of the proofs which were left as exercises have been fleshed out. Thesenotesarenotofficiallyendorsedbythethreelecturersandarearemerelythescribers own notes. Contents 0 Background Material 1 0.1 Algebras and Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.2 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 Ordinary Representation Theory 7 1.1 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Ordinary Representation Theory . . . . . . . . . . . . . . . . . . . . . . 10 1.4 The Ordinary Character Table . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Semisimple Group Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.7 Integrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.8 Representations and Subgroups . . . . . . . . . . . . . . . . . . . . . . . 19 2 Modular Representation Theory 21 2.1 Modular Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Change of Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Brauer Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Grothendieck Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5 The Decomposition Map . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6 Projective FG-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.7 Projective OG-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.8 p-Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.9 Relative Projectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.10 Vertices and Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.11 The Green Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.12 Sources of Simple Modules . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.13 Endopermutation Modules Over p-Groups . . . . . . . . . . . . . . . . . 29 2.14 The Dade Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.15 The Green Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3 Block Theory 33 3.1 Measuring Semisimplicity . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Module Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 Twisted Group Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 b-Brauer Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Brauer’s First Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . 40 i Contents ii 3.6 Brauer’s Third Main Theorem . . . . . . . . . . . . . . . . . . . . . . . 41 3.7 Fusion System of a Block . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.8 Structure of Centric Brauer Pairs . . . . . . . . . . . . . . . . . . . . . . 42 3.9 Known Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.10 Alperin’s Weight Conjecture . . . . . . . . . . . . . . . . . . . . . . . . 43 3.11 Broué’s Abelian Defect Group Conjecture . . . . . . . . . . . . . . . . . 44 3.12 Finiteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.13 Weak Donovan Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.14 Blocks in Characteristic 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.15 Blocks and the cde Triangle . . . . . . . . . . . . . . . . . . . . . . . . 48 3.16 Brauer’s Second Main Theorem . . . . . . . . . . . . . . . . . . . . . . 49 3.17 Block Orthogonality Relations . . . . . . . . . . . . . . . . . . . . . . . 49 3.18 Characters and Morita Equivalence . . . . . . . . . . . . . . . . . . . . . 50 0. Background Material In this section we will gather together some basic background material that may be unfa- miliar. We also outline the notation that we will use for the remainder of these notes. 0.1 Algebras and Modules Let F be a field, then we recall the definition of an algebra over a field. Definition 0.1.1. Let A be a vector space over F then A is an F-algebra if A has a binary operation · : A×A → A such that for all x,y,z ∈ A and a,b ∈ F we have • (x +y)·z = x ·z +y ·z, • x ·(y +z) = x ·y +x ·z, • (ax)·(by) = (ab)(x ·y), • x ·(y ·z) = (x ·y)·z, • there exists 1 ∈ A such that 1 ·x = x = x ·1 . A A A In our definition of F-algebra we have assumed that it is associative and has a unit element. Note that in particular A is a ring and hence we have the following definition of an A-module. Definition 0.1.2. A left A-module is an abelian group M, (whose group operation we denote additively), together with an operation · : A×M → M such that, for all a,b ∈ A and m,n ∈ M we have • a·(m+n) = a·m+a·n, • (a+b)·m = a·m+b·m, • (ab)·m = a·(b·m), • 1 ·m = m. A We also say a subgroup N (cid:54) M is a left A-submodule if for all a ∈ A, n ∈ N we have a·n ∈ N. Section 0.1 2 Remark 0.1.3. We can define modules for an arbitrary ring but here we have chosen to define them only for F-algebra’s as this is the only context in which we will need them. As any F-algebra contains a copy of F we have any A-module is in fact an F-vector space. In light of this we will refer to the dimension of M, sometimes denoted dimM or dim (M), F as the dimension of M as an F-vector space. Remark 0.1.4. We can also define right A-modules and right A-submodules in an analo- gous way. All A-modules and submodules are assumed to be left A-modules and submodules unless otherwise stated. If we wish to emphasise whether M is a left, (resp. right), A-module then we will write M as M, (resp. M ). A A Let M be an A-module and N (cid:54) M a submodule of M. We can form the quotient group M/N and indeed this is again an A-module called the quotient module. Definition 0.1.5. Let A, B be two F-algebras. We say M is an (A,B)-bimodule if M is a left A-module and a right B-module such that for all a ∈ A, b ∈ B and m ∈ M we have a·(m·b) = (a·m)·b. We will emphasise this bimodule structure with the notation M . A B If M and N are A-modules then we write Hom (M,N) for the abelian group of all A group homomorphisms f : M → N, which respect the action of A. In other words we have f(a·m) = a·f(m) for all m ∈ M, a ∈ A. Furthermore we denote Hom (M,M) by A End (M) the group of all endomorphisms of M. A Definition 0.1.6. Assume A is an F-algebra and M is an A-module. We say M is finitely generated if there exists a finite subset {m ,...,m } ⊆ M such that every element of M 1 s is an A-linear combination of the m , (in other words M = (cid:80)s Am ). i i=1 i We will assume that all A-modules are finitely generated unless specifically stated otherwise. In particular A itself is finitely generated. We recall the following fundamental result concerning finitely generated modules. Proposition 0.1.7. An A-module M is finitely generated if and only if for every ascending chain M (cid:54) M (cid:54) M (cid:54) ··· 1 2 3 of submodules M (cid:54) M terminates. In other words there exists an index j such that i M = M for all k (cid:62) j. j k Let us now introduce a collection of A-modules which will be the building blocks for all other modules. Definition 0.1.8. An A-module S is called simple if S is non-zero and the only submodules of S are {0} and S. Furthermore an A-module M has a composition series if there exists a finite series of submodules {0} = M < M < ··· < M < M = M 0 1 k−1 k Section 0.1 3 such that the quotient modules M /M are simple for all 1 (cid:54) i (cid:54) k. The modules i i−1 {M /M | 1 (cid:54) i (cid:54) s} are called the factors of the composition series and k is called the i i−1 length of the composition series. Let us recall some basic results on modules, which are analogues of the usual isomorphism theorems for groups and rings. These are left as an easy exercise for the reader. Theorem 0.1.9. Let A be an F-algebra and M and N two A-modules. (i) Assume f : M → N is an A-module homomorphism then Ker(f) and Im(f) are submodules and we have an isomorphism M/Ker(f) → Im(f). (ii) Assume M and N are submodules of a common A-module then we have an isomor- ∼ phism (M +N)/M = N/(M ∩N). (iii) Assume N is a submodule of M. There exists a one to one inclusion-preserving correspondence, P (cid:55)→ P/N, between submodules P of M which contain N and submodules of M/N. The existence of a composition series for a finitely generated A-module follows easily from part (iii) of Theorem 0.1.9 and Proposition 0.1.7. There is a small caveat to the existence of a composition series in that many composition series may exist for a module. However the next result says at least that the length and factors are independent of the choice of a composition series. We call the common length of a composition series the composition length of M and the factors the composition factors of M. Theorem 0.1.10 (Jordan-Hölder). Suppose that an A-module M has two composition series {0} = M < M < ··· < M = M, 0 1 k {0} = N < N < ··· < N = N, 0 1 (cid:96) then there exists a bijection σ : {M /M ,...,M /M } → {N /N ,...,N /N } 1 0 k k−1 1 0 (cid:96) (cid:96)−1 such that X and σ(X) are isomorphic for all X ∈ {M /M ,...,M /M }. 1 0 k k−1 Proof. We prove this by induction on the length of the composition series. Any module having a composition series of length 1 must be simple and the theorem is trivially true for simple modules. Let us now assume the theorem is true for all modules which have a composition series whose length is less than k. There are two main cases to consider. Firstly assume M = N then removing M from the composition series gives a com- k−1 (cid:96)−1 position series for M = N . By assumption we have k − 1 = (cid:96) − 1 ⇒ k = (cid:96) and k−1 (cid:96)−1 M /M ∼= N /N for all 1 (cid:54) i (cid:54) k − 1. Finally as M = N we clearly have i i−1 i i−1 k−1 (cid:96)−1 M/M = M/N so the theorem is proved. k−1 (cid:96)−1 Section 0.2 4 Now assume that M (cid:54)= N then we will consider the following submodule V = k−1 (cid:96)−1 M ∩N of M. As M (cid:54)= N we must have that V is a proper submodule of either k−1 (cid:96)−1 k−1 (cid:96)−1 M or N . Let us assume without loss of generality that M ∩ N is a proper k−1 (cid:96)−1 k−1 (cid:96)−1 submodule of M , in particular N (cid:54)⊂ M . By (iii) of Theorem 0.1.9 we have M k−1 (cid:96)−1 k−1 k−1 is a maximal submodule of M because the quotient M/M is simple. Therefore we have k−1 M ⊆ M +N ⊆ M which implies M +N ⊆ M because N (cid:54)⊂ M . By (ii) k−1 k−1 (cid:96)−1 k−1 (cid:96)−1 (cid:96)−1 k−1 of Theorem 0.1.9 we have isomorphisms M/M ∼= N /(M ∩N ) M/N ∼= M /(M ∩N ), (0.1) k−1 (cid:96)−1 k−1 (cid:96)−1 (cid:96)−1 k−1 k−1 (cid:96)−1 in particular the quotient modules N /(M ∩N ) and M /(M ∩N ) are simple. (cid:96)−1 k−1 (cid:96)−1 k−1 k−1 (cid:96)−1 Let {0} = L < L < ··· < L < L = M ∩ N be a composition series for 0 1 s−1 s k−1 (cid:96)−1 M ∩N then by Eq. (0.1) we have composition series k−1 (cid:96)−1 {0} = L < L < ··· < L < M ∩N < M , 0 1 s−1 k−1 (cid:96)−1 k−1 {0} = L < L < ··· < L < M ∩N < N 0 1 s−1 k−1 (cid:96)−1 (cid:96)−1 of M and N . We may now use the induction hypothesis because M and N k−1 (cid:96)−1 k−1 (cid:96)−1 clearly have composition series whose length is less than k. Using this we have the above composition series must have the same length as the composition series {0} = M < M < ··· < M < M , 0 1 k−2 k−1 {0} = N < N < ··· < N < N . 0 1 (cid:96)−2 (cid:96)−1 In particular we have k = (cid:96). Now again by Eq. (0.1) we have the following are composition series {0} = L < L < ··· < L < M ∩N < M < M , 0 1 s−1 k−1 k−1 k−1 k {0} = L < L < ··· < L < M ∩N < N < N 0 1 s−1 k−1 k−1 k−1 k of M. The theorem then follows by comparing these composition series with the original composition series using the induction hypothesis and Eq. (0.1). (cid:4) 0.2 Categories In this section we will recall some of the fundamental terminology used in the language of categories. We will not recall the notions of triangulated or derived categorories but instead refer the reader to [Wei94, Chapter 10]. Definition 0.2.1. A category C consists of a class of objects Obj(C) and a set of morphisms Hom (M,N) defined for each pair of objects M,N ∈ Obj(C). For each object M ∈ C Obj(C) there must exist an identity morphism Id ∈ Hom (M,M) and for any three ob- M C jects L,M,N ∈ C there must exist a composition function Hom (L,M)×Hom (M,N) → C C Hom (L,N). These composition functions must satisfy C Section 0.2 5 • for all f : K → L, g : L → M and h : M → N we have (hg)f = h(gf) • for all f : M → N we have Id f = f = f Id . N M Example 0.2.2. The standard example of a category Sets whose objects are sets and whose morphisms are just set functions. Composition is then just given by the usual compositionofsetfunctions. Onecanalsoformthecategoryofgroups(denotedGroups), rings (denoted Rings) and finitely generated A-modules (denoted A–mod) where A is an F-algebra. Remark 0.2.3. We will drop the notation Obj if it is clear that we are talking about an object in the given category, i.e. we may write M ∈ A–mod to mean M is a finitely generated A-module. Definition 0.2.4. If C is a category then a morphism f ∈ Hom (M,N) is an isomorphism C if there exists a morphism g ∈ Hom (N,M) such that gf = Id and fg = Id . C M N Example 0.2.5. An isomorphism in Sets is just a set bijection. In Groups and Rings this term has its usual meaning. Definition 0.2.6. If C and D are two categories then a functor F : C → D is a rule that associates to every object M ∈ Obj(C) an object F(C) ∈ Obj(D) and to every morphism f ∈ Hom (M,N) a morphism F(f) ∈ Hom (F(M),F(N)) such that F(Id ) = Id C D M F(M) and F(gf) = F(g)F(f) for all f ∈ Hom (L,M) and g ∈ Hom (M,N). C C Example 0.2.7. We have a functor F : Groups → Sets called the forgetful functor. It is such that for any group G ∈ Groups we have F(G) is simply G as a set with the additional binary operation forgotten. Similarly any morphism f ∈ Hom(G,H) is such that F(f) is the underlying set function with the additional homomorphism property forgotten. One can define a forgetful functor for any category whose objects are sets. In representation theory we often want to know when A–mod and B–mod are equivalent where A and B are two F-algebras. We now make this notion precise on a categorical level. Definition 0.2.8. Assume C and D are two categories and F : C → D and G : C → D are two functors. A natural transformation η : F ⇒ G is a rule that associates a morphism η(M) ∈ Hom (F(M),G(M)) such that for every morphism f ∈ Hom (M,M(cid:48)) D C the following diagram commutes F(f) F(M) F(M(cid:48)) η(M) η(M(cid:48)) G(M) G(M(cid:48)) G(f) If η(M) is an isomorphism for each M ∈ Obj(C) we call η a natural isomorphism. Section 0.2 6 Example 0.2.9. Let A be an F-algebra and M ∈ A–mod a finitely generated A-module. We have a functor M ⊗ − : A–mod → A–mod such that for any finitely generated A A-module N we have (M ⊗ −)(N) = M ⊗ N and for any morphism f ∈ Hom (N,N(cid:48)) A A A we define (M ⊗ −)(f) to be the morphism given by A (M ⊗ −)(f)(m⊗n) = m⊗f(n) A for all m ⊗ n ∈ M ⊗ N. Assume now that g ∈ Hom(M,M(cid:48)) is a morphism of finitely A generated A-modules then we have a natural transformation g⊗− : M⊗ − ⇒ M(cid:48)⊗ −. A A This is such that for any finitely generated A-module N we have (g ⊗−)(N) = g ⊗N ∈ Hom(M ⊗ N,M(cid:48) ⊗ N) is such that A A (g ⊗N)(m⊗n) = g(m)⊗n for all m⊗n ∈ M ⊗ N. A Definition 0.2.10. If C and D are categories then a functor F : C → D is an equivalence of categories if there is a functor G : D → C and natural isomorphisms Id ⇒ GF C and Id ⇒ FG. Here Id denotes the identity functor on C, i.e. the functor such that D C Id (M) = M for all M ∈ Obj(C) and Id (f) = f for all morphisms f ∈ Hom (M,N). C C C Using the language of categories we can now give one precise meaning to the notion that two F-algebras have the same “representation theory”. There are other notions which encapsulate different levels of information Definition 0.2.11. Assume A and B are two F-algebras. We say A and B are Morita equivalent if there exists an (A,B)-bimodule P and a (B,A)-bimodule Q such that P ⊗ B Q ∼= A as (A,A)-bimodules and Q⊗ P ∼= B as (B,B)-bimodules. If this is the case then A the functors Q⊗ − : A–mod → B–mod and P ⊗ − : B–mod → A–mod are mututally A B inverse equivalences of categories. Remark 0.2.12. In the above definition one can replace A and B by arbitrary rings and A–mod and B–mod by the full module categories.

An Introduction to the Representation Theory of Finite Groups

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