Representation theory of finite groups

Robert Hartmann

May 12, 2003

Abstract

This mini course is intended as a brief introduction to the representation theory of finite groups, with a focus on the modular theory. No proofs will be given, and technical details will be omitted whenever possible. The main purpose is to introduce some recent open problems.

1 Introduction

Let G be a finite group, and let k be a commutative ring. We are interested in modules over the group algebra kG = {∑_{g∈G} α_{g}g | α_{g} ∈ k}. However, the representation theory of kG depends heavily on the base ring k. We will need to make further assumptions on k, and the cases we are interested here are given in the following definition.

1.1 Definition Let p be a prime number. A p-modular system (K, O, F) for G is a triple consisting of a complete discrete valuation ring O with maximal ideal ℘, its quotient field K and its residue field F = O/℘ such that F has characteristic p, K has characterstic zero, and K is large enough for G, e.g., K contains a primitive exp(G)-th root of unity. We will also assume that F is algebraically closed, though this will not be necessary for some of the results. We can assume that K contains the rational number, and O contains the integers.

One might think of the following typical example for a p-modular system. Start with the field ℚ_{p} (the completion of the rationals with respect to the padic topology),adjoin a primitive exp(G)-th root of unity and all l-th roots of unity, where l runs through the set of primes different from p. This will be the field K. Next define O to be the integral closure of ℤ_{p} in K, and finally let F be the quotient of O by its unique maximal ideal. Then F will be an algebraic closure of the field with p elements.

The representation theory of KG is the easiest, it is essentially the same as for ℂG and usually called ordinary representation theory. This is because KG is semisimple by Maschke’s theorem, and there is only a finite number of isomorphism classes of indecomposable KG-modules. These are in one-to-one correspondence with the irreducible characters of G.

On the other hand, if the characterstic p of F divides the order of the group G (which we will always assume) then F G is not semisimple, and only for very few groups it is possible to classify all indecomposable F G-modules up to isomorphism (we will be more precise later on). The representation theory of F G is called modular theory.

Replacing F by the ring O, we get to the integral representation theory, and things get even more difficult. Instead of modules we consider now only OG-lattices, i.e., modules which are finitely generated and free over O. For these, we at least have the notion of a representation, while in general for an OG-lattice it is not possible to choose an O-basis.

The ring O, besides being interesting by itself, provides a link between the ordinary and the modular theory, as follows. Starting with an OG-lattice V, we can extend scalars and get the KG-module K ⊗_{O} V. In this way it is also possible to assign a character to V, though this does not determine the lattice up to isomorphism. On the other hand, we can consider F ⊗_{O} V, or isomorphically V/℘V, which is an F G-module. In fact, many results hold for OG- lattices as well as for F G-modules, and in the future, whenever we write R, we mean either F or O.

Most importantly, by our assumptions the following crucial theorem holds for RG-lattices.

1.2 Theorem (Krull-Schmidt) Let V be any RG-lattice, and let V = V_{1} ⊕ ⋯ ⊕ V_{n} and V = W_{1} ⊕ ⋯ ⊕ W_{m} be two decompositions of V into indecomposable RG- lattices. Then n = m and there exists a permutation π such that V_{π(i)} ≅ W_{i}.

Thus, to understand an RG-module, it is enough to examine its indecomposable summands. Another major theorem is the following.

1.3 Theorem (Greens indecomposability theorem) Let H be a normal subgroup of G of index p, and let V be an indecomposable RG-lattice. Then also the induced lattice Ind_{H}^{G}(V) = RG ⊗_{RH} V is indecomposable.

The theorem has many important consequences. We just mention its application to p-groups.

1.4 Corollary Let P be a p-group, Q any subgroup of P, and let V be an indecomposable RG-lattice. Then Ind_{Q}^{P}(V) is indecomposable. In particular, the regular RG-lattice RG is indecomposable, and it is the only indecomposable projective RG-lattice.

2 Brauer characters and the cde triangle

Denote by μ(R) the group of roots of unity of the ring R. Since O is integrally closed, we have μ(O) = μ(K). On the other hand, there exists an epimorphism μ(O) → μ(F) with kernel consisting of all elements with order a power of p. Thus we can identify the roots of unity ϵ of p′-order in O with the roots of unity \overline{ϵ} in F, and call ϵ ∈ μ(O)_{p′} the lift of \overline{ϵ} ∈ μ(F).

Now let V be any F G-module, and let Δ : G → GL_{n}(F) be a corresponding representation. For g ∈ G, let \overline{λ}_{1}, …, \overline{λ}_{n} be the eigenvalues of Δ(g), and set

φ_{M}(g) := λ_{1} + … λ_{n}

. It is an easy exercise to show that if g = g_{p}g_{p′} is the decomposition of g into its p- and its p′-part, then φ_{M}(g) = φ_{M}(g_{p′}).

2.1 Definition The restriction

φ_{M} : G_{p′} → K

is called the Brauer character of M.

Brauer characters are class functions, i.e., constant on conjugacy classes of G. Clearly the Brauer characters have values in the field ℚ(ζ), where ζ denotes an exp(G)-th root of unity. Thus we can view the Brauer character of M as a class function of G_{p′} with values in the complex numbers.

2.2 Theorem

(i) If 0 → L → M → N → 0 is a short exact sequence of F G-modules then φ_{M} = φ_{L} + φ_{N}. (ii) One has φ_{M⊗_{F} N} = φ_{M} · φ_{N}, where the product on the right hand side is pointwise multiplication. (iii) If S_{1}, …, S_{r} are representatives for the isomorphism classes of simple F Gmodules, then their Brauer characters φ_{1}, …, φ_{r} form a ℂ-basis of the vector space of class functions of G with values in ℂ which vanish outside of G_{p′}.

The Brauer characters of the simple F G-modules are called the irreducible Brauer characters, and we write IBr(G) = {φ_{1}, …, φ_{r}}. Let R_{F} (G) denote the set of integral linear combinations of IBr(G). Then R_{F} (G) is isomorphic to the Grothendieck group of the category _{F G}mod of F G-modules. This group is constructed by considering the free abelian group on all isomorphism classes of F G-modules and factoring out the ideal generated by the elements [N]−[M]−[L] for each short exact sequence 0 → M → N → L → 0.

The number of irreducible Brauer characters is just the number of p′-conjugacy classes of G. The values of the irreducible Brauer characters on the p′-conjugacy classes are usually represented in a matrix, called the Brauer character table.

Denote by R_{K}(G) the ring of K-valued characters of G (or the Grothendieck group of _{KG}mod).

2.3 Definition We define the decomposition map

d_{G} : R_{K}(G) → R_{F} (G)

as follows. If M is any KG-module then there exists an OG-lattice L such that K ⊗_{O} L ≅ M. Such a lattice V is called an O-form of M. Note that M can have many non-isomorphic O-forms. However, if one considers the F G-module \overline{V}, then the composition factors of \overline{V} are (up to isomorphism and ordering) uniquely determined by M. We define d_{G}([M]) to be the sum of the classes of these composition factors (counted with multiplicities).

By the above, we can view d_{G} as a map between the rings of ordinary characters and Brauer characters. Then d_{G} is given just by restriction from G to G_{p′}. The representing matrix for d_{G} indexed by Irr(G) × IBr(G) is called the decomposition matrix, and is usually denoted by D.

2.4 Theorem The decomposition map d_{G} is a surjective ring homomorphism, and it commutes with conjugations, restrictions and inductions (i.e., it is a morphism of Mackey functors).

Over KG, the notions of simple and indecomposable modules coincide, hence the character ring R_{K}(G) decodes information about both kinds of modules. However, over F G or OG, the ring of Brauer characters R_{F} (G) stores only information about simple modules, and there is yet another ring that is generated by isomorphism classes of indecomposable modules.

2.5 Definition Let a_{R}(G) be the ring with generators the isomorphism classes [M] of RG-lattices, and relations

[M] + [N] = [M ⊕ N] and [M] · [N] = [M ⊗ N].

The identity and zero elements are given by the classes of the trivial and the zero module. We call a_{R}(G) the representation ring of RG. For R = F, a(G) = a_{F} (G) is also called the Green ring. As a subring, a_{R}(G) contains the ring P_{R}(G) which is generated by the isomorphism classes of projective RG-lattices.

Note that in general a_{R}(G) is not finitely generated. Very little can be said about the structure of a(G) in general.

We define the Cartan map c_{G} : P_{F} (G) → R_{F} (G) by assigning to the class of a projective indecomposable F G-module the sum of the classes of its composition factors. Note that the representing matrix C of c_{G} is a square matrix, indexed by the isomorphism classes of simple resp. projective indecomposable modules.

2.6 Theorem The Cartan map is injective. More precisely, if |G| = mp^{a} with p |̸ m, then p^{a} · R_{F} (G) ⊆ c_{G}(P_{F} (G)) and p^{a−1} · R_{F} (G) ⊈ c_{G}(P_{F} (G)). In particular, the determinant of the Cartan matrix is a power of p.

Now we complete the cde-triangle by defining a map e_{G} : P_{F} (G) → R_{K}(G). For a projective F G-module P there exists up to isomorphism a unique projective OG-lattice P_{0} such that F ⊗_{O} P_{0} ≅ P, and we can extend scalars to obtain a KG-module M := K ⊗_{O} P_{0}. Now we define e_{G}([P]) := [M] ∈ R_{K}(G). It is clear from the construction that d_{G}([M]) is the sum of the classes of the composition factors of P in R_{F} (G) and hence d_{G} ∘ e_{G} = c_{G}, and this identity can be pictured in a commutative triangle diagram.

2.7 Theorem The image of e_{G} in R_{K}(G) consists of all χ which vanish on the p-singular elements of G.

2.8 Example We consider the two extreme cases:

(i) If p |̸ |G| then R_{K}(G), R_{F} (G) and P_{F} (G) are all isomorphic, and the matrices corresponding to c_{G}, d_{G} and e_{G} are all the identity matrices. (ii) If G is a p-group, then R_{F} (G) and P_{F} (G) have rank one, corresponing to the trivial and the regular module. Thus the decomposition matrix is the 1 × 1matrix with entry |G|. The matrix E describing e_{G} consists of one row, indexed by Irr(G), and its entries are the multiplicities of the irreducible characters in the regular module KG, hence the character degrees. The decomposition matrix consists of one column, also containing the character degrees. Hence D = E^{T}, which is in fact true in general.

2.9 Theorem One has D = E^{T}, and hence C = D ·D^{T} is a symmetric matrix.

2.10 Example

Let G = Σ_{4} be the symmetric group on four letters. First we recall the ordinary character of G:

1 (12) (12)(34) (123) (1234)
1 1 1  1  1  1
sgn 1 −1  1  1  −1
χ_{2} 2 0  2  −1  0
χ_{3} 3 1  −1  0  −1
χ′_{3} 3 −1  −1  0  1

For the prime p = 2, the Brauer characters will be non-zero only on the classes of 1 and (123). The two one dimensional characters obviously have the same decomposition and correspond to the trivial Brauer character. Moreover, the trivial character is one the only Brauer character of degree one, since the simple F Gmodules of dimension one correspond to the elements of the group Hom(G, F^{×}), and for F of characteristic 2 this group is trivial. Next, χ_{2} corresponds to an F G-module of degree two, which must be simple, since it cannot have two trivial composition factors bcause of χ_{2}(123) = −1 ≠ 2 = 2 · 1(123). Hence we have found all simple F G-modules, and the Brauer character table is given by

\begin{aligned}& 1 & (123) \\ 1 & 1 & 1 \\ φ & 2 & −1 \end{aligned}

and the decomposition matrix is

D = ( \begin{matrix}1 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 \end{matrix} )

The Cartan matrix can now be computed as DD^{T}, and the case of p = 3 is left as an exercise.

Next we describe the probably most famous open conjecture in the representation theory of finite groups, the so-called weight conjecture by J.Alperin dating back to 1987. Following Alperin, we call a pair (P, [S]) a weight of G, where P is a p-subgroup of G and [S] is an isomorphism class of simple projective F (N_{G}(P)/P)-modules. Note that G acts on the set of all weights for G by conjugation, since for x ∈ G, ^{x}P is a p-subgroup, and ^{x}S is a simple projective F (N_{G}(^{x}P)/^{x}P)-module. Write S_{p}(G) for the set of all p-subgroups of G (including the trivial subgroup), and denote by l(G) the number of simple F G-modules, and by l_{0}(G) the number of simple projective F G-modules.

2.11 Conjecture (Alperin’s weight conjecture, 1987) The number l(G) of simple F G-modules equals the number of G-orbits of weights of G, i.e.,

l(G) = ∑_{P ∈S_{p}(G)/G} l_{0}(N_{G}(P)/P).

If we denote by S_{p}^{∗}(G) the set of non-trivial p-subgroups of G the the conjecture is obviously equivalent to

l(G) − l_{o}(G) = ∑_{P ∈S_{p}^{∗}(G)/G} l_{0}(N_{G}(P)/P),

and the subgroups appearing on the right hand side of the equation are all of strictly smaller order than G. We will later encounter several refinements of this conjecture. Exercise: Verify Alerin’s conjecture for G = Σ_{4} and p = 2, 3.

3 Vertices and Sources, the Green correspondence and trivial source modules

For two RG-lattices M and N we write M|N to denote that M is isomorphic to a direct summand of N.

Let H be a subgroup of G. We say that the RG-lattice M is projective relative to H ( or just H-projective) if the equivalent conditions in the following theorem are satisfied.

3.1 Theorem For an RG-lattice M and a subgroup H of G the following are equivalent:

(i) M|Ind_{H}^{G}(X) for some RH-lattice X. (ii) M|Ind_{H}^{G}(Res_{H}^{G}(M)). (iii) Every short exact sequence 0 → L → N → M → 0 of RG-lattices, which splits as a sequence of RH-lattices, also splits as a sequence of RG-lattices. (iv) Let f : M → L and g : N → N be homomorphism of RG-lattices such that g is surjective. If there exists an RH-lattice homomorphism h : M → N such that g ∘ h = f then there exists also an RG-lattice homomorphism with that property.

(v) (Higman’s criterion) There exists f ∈ End_{RH}(M) such that 1_{M} = ∑_{g∈G/H} ^{g}f, where ^{g}f ∈ End_{R}(M) is defined by (^{g}f)(m) = g · f(g^{−1} · m) for m ∈ M.

Note that M being projective relative to the trivial subgroup means just that M is a projective RG-lattice. Moreover, if the index [G : H] is not divisible by p then every RG-lattice is H-projective. This can be seen by using Higman’s criterion with f = [G : H]^{−1} · 1_{M}.

If M is H-projective, and H ≤ K ≤ G then M is also K-projective. Note also that for a H-projective OG-lattice V, the F G-module \overline{V} = F ⊗_{O} V is always H-projective, but the converse is not true.

3.2 Theorem Let M be an indecomposable RG-lattice. Then there exists a minimal subgroup Q of G such that M is projective relative to Q. Moreover, Q is always a p-subgroup, and any two minimal subgroups with this property are conjugate under G.

We call the subgroups Q the vertices of M. If Q is a vertex of M then by condition (i) for Q-projectivity, there exists an indecomposable RQ-lattice S such that M|Ind_{Q}^{G}(S). We call S a source of M.

3.3 Theorem (Green correspondence) Let P ≤ G be a p-subgroup, and set N := N_{G}(P).

(i) Let M be an indecomposable RG-lattice with vertex P. Then Res_{N}^{G}(M) has up to isomorphism a unique direct summand L with vertex P, which occurs with mutiplicity one, and all other direct summands have vertices strictly smaller then P. (ii) Let L be an indecomposable RN-lattice with vertex P. Then Ind_{N}^{G}(L) has up to isomorphism a unique direct summand M with vertex P, which occurs with mutiplicity one, and all other direct summands have vertices strictly smaller then P. (iii) This defines mutually inverse bijections between the sets of isomorphism classes of indecomposable RG- and RN-lattices with vertex P.

3.4 Definition Let X be a finite G-set (i.e. a finite set equipped with an action of G). Then we can form the free R-module with basis X, and the action of G on X extends to an RG-module structure. We call RX a permutation RG-module.

It is an easy exercise to show that if X is a transitive G-set (i.e., consists of just one G-orbit), then RX ≅ Ind_{H}^{G}(R) where H = Stab_{G}(x) for any element x ∈ X. Obviously RX is H-projective. If G is a p-group then RX is indecomposable, and H is a vertex of RX. In general, however, the module Ind_{H}^{G}(R) might not be indecomposable. Hence, instead of permutation modules, one often considers the following more general class of modules.

3.5 Definition An RG-lattice M is called a p-permutation RG-module (or trivial source RG-module) if the following equivalent conditions hold.

(i) M is a direct summand of a permutation RG-module. (ii) The restriction of M to a Sylow p-subgroup of G is a permutation module. (iii) The source of every indecomposable summand of M is the trivial module.

Note that there are only finitely many isomorphism classes of p-permutation RG-modules. If M and N are p-permutation RG-modules then so are M⊕N and M ⊗ N, which shows that the isomorphism classes of p-permutation modules form a subring T_{R}(G) of the representation ring a_{R}(G). One can also show that the Green correspondece preserves sources, hence p-permutation modules correspond to p-permutation modules under the bijections given by the Green correspondence.

4 Blocks of group algebras

Recall that an idempotent e of an R-algebra A is a non-zero element e ∈ A satisfying e^{2} = e. Clearly 1 is always an idempotent. We call a set {e_{i}}_{i∈I} of idempotents (pairwise) orthogonal if e_{i}e_{j} = 0 whenever i ≠ j. An idempotent e is called primitive if it is not possible to write e = e_{1} + e_{2} for orthogonal idempotents e_{1} and e_{2}. In an R-algebra A one can always write 1 = e_{1} + ⋯ + e_{n} for primitive orthogonal idempotents e_{i} (1 ≤ i ≤ n), and any two such decompositions are conjugate in A.

Now consider the R-algebra Z(RG), and write 1 = b_{1} + … b_{n} for primitive orthogonal idempotents b_{1}, …, b_{n}. Since the b_{i} are central in F G, one has b_{i}RG = b_{i}RGb_{i} = RGb_{i}, and if we set B_{i} = b_{i}RG it follows that RG = B_{1} ⊕ ⋯ ⊕ B_{n} is a decomposition of RG into two sided ideals, and each B_{i} is an R-algebra on itself. We call the B_{i} the blocks of RG, and the b_{i} are called the block idempotents. If M is any RG-lattice then M = b_{1}M ⊕ ⋯ ⊕ b_{n}M. In particular, if M is indecomposable then there exists a unique block B_{i} such that b_{i}M ≠ 0, and we say that M belongs to the block B_{i}. In the same way we can associate a block to a simple RG-lattice and to a Brauer character.

If e ∈ Z(OG) is a primitive idempotent, then the element ē = 1 ⊗_{O} e ∈ F ⊗_{O} F G is a central primitive idempotent of F G. Conversely, one can show that any primitive idempotent ē of Z(F G) can be lifted uniquely to a central primitive idempotent of Z(OG). Hence we obtain a one-to-one correspondence of blocks of OG and F G.

If e is a block idempotent in Z(OG) then e is also a central idempotent of KG (though in general not primitive). Hence for any simple KG-module V, there exisists a unique block B_{i} of OG such that b_{i}V ≠ 0, and we say that that V and its irreducible character χ_{V} lie in the block B_{i}.

The block that contains the trivial character is called the prinicipal block of OG.

4.1 Theorem (i) Two irreducible characters χ and χ′ lie in the same block of OG if and only if for every x ∈ G

\frac{|C_{x}|χ(x)}{χ(1)} ≡ \frac{|C_{x}|χ′(x)}{χ′(1)} mod ℘,

where C_{x} denotes the conjugacy class of x in G. Note that both expressions are in ℤ[ζ] for ζ an exp(G)-th root of unity. (ii) Two simple F G-modules S_{i} and S_{j} ( and their irreducible Brauer characters φ_{i} and φ_{j}) lie in the same block of OG if and only if there is a list of indices i = i_{0}, i_{1}, …, i_{n} = j such that S_{i_{l−1}} is a composition factor of P_{i_{l}}, where S_{1}, …, S_{r} is a complete list of simple F G-modules, and P_{l} is a projective cover of S_{l}.

Part (ii) of the theorem just says that the blocks of OG correspond to the best possible decomposition of the Cartan matrix as a block diagonal matrix. This also allows us to talk about the decompoition matrix of a block, or the Cartan matrix of a block.

We van view the group algebra RG as an R[G × G]-module by

(g, h) · x = gxh^{−1}

for g, h ∈ G and x ∈ RG. Submodules of RG with respect to this module structure are just two sided ideals of RG. Thus the indecomposable submodules are exactly the blocks. Let ΔG be the diagonal subgroup of G × G, i.e., the subgroup whose elements are all (g, g) for g ∈ G. There is an isomorphism of R[G × G]-modules

Ind_{ΔG}^{G×G}(R) = R[G × G] ⊗_{R[ΔG]} R → RG

given by

(g, h) ⊗_{R[ΔG]} α ↦ αgh^{−1}.

Hence RG is projective relative to ΔG, and so are the indecomposable summands of RG, which are the blocks. So the vertices of a block (considered as an R[G × G]-module) are p-subgroups of ΔG. But obviously ΔG is isomorphic to G, so these vertices are of the form ΔD for a p-subgroup D of G. We call such a D the defect group of the block. If |D| = p^{d} then we say that the block has defect d, and write d(B) for the defect of the block B. One can show that the definition of defect groups does not depend on whether one works over F or O. There are several equivalent characterizations of defect groups, two of them we will give below.

Let A = F G, now viewed as an F G-module with the G-action given by conjugation, i.e., ^{g}a = gag^{−1} for g ∈ G and a ∈ A. Let H ≤ G, and denote by A^{H} the H-fixed points of A, i.e., the set of all a ∈ A such that ^{h}a = a for all h ∈ H. For subgroups U ≤ H ≤ G we define the (relative) trace map

tr_{U}^{H} : A^{U} → A^{H}, a ↦ ∑_{h∈H/U} ^{h}a.

The the defect groups of a block B are precisely the minimal subgroups D of G such that 1_{B} is in the image of tr_{D}^{G}.

A third way to define defect groups is the following. Let S_{1}, …, S_{t} be those simple F G-modules which lie in the block B. Then among the set of conjugacy classes of p-subgroups which occur as vertices of S_{1}, …, S_{t} there is a unique maximal class, and these are exactly the defect groups of B.

Not all p-subgroups of G can occur as defect groups of blocks. One can show that every defect group can be written as P ∩^{x}P for a Sylow p-subgroup P of G. In particular, the intersection of all Sylow p-subgroups O_{p}(G) must be contained in every defect group.

We write Irr(B) and IBr(B) for the set of all ordinary and Brauer characters that lie in the block B. Then the defect of B can be computed as

d(B) = min{d ≤ 0 | χ(1)_{p} · p^{d} ≥ |G|_{p} for all χ ∈ Irr(B)}

or as

d(B) = min{d ≤ 0 | φ(1)_{p} · p^{d} ≥ |G|_{p} for all φ ∈ IBr(B)}.

For χ ∈ Irr(G) one defines the defect d(χ) of χ by χ(1)_{p} · p^{d(χ)} = |G|_{p}. Thus the defect of the block B is the maximum of the defects of all characters that lie in B.

Of particular interest are the blocks of defect zero.

4.2 Theorem Let B be a block of F G. The following are equivalent

(i) d(B) = 0. (ii) There is an irreducible character χ in B such that χ(1)_{p} = |G|_{p}. (iii) There is exactly one irreducible character in B. (iv) The decomposition matrix of B equals (1).

(v) The Cartan matrix of B equals (1). (vi) There is a simple projective F G-module in B (vii) B is isomorphic to a matrix algebra M_{n×n}(F).

There is another notion closely related to blocks.

4.3 Definition A central character of F G is an F -algebra epimorphism

θ : Z(F G) → F.

Let b_{1}, …, b_{n} be the block idempotents of F G. Then there are central characters θ_{i} : Z(F G) → F given by

θ_{i}(b_{i}) = 1 and θ_{i}(b_{j}) = 0 for i ≠ j,

and every central character arises this way. Thus the blocks are in one-to-one correspondence with the central characters.

Now let P be a p-subgroup of G. We define the Brauer homomorphism

σ_{P} : Z(F G) → Z(F [C_{G}(P)]), ∑_{g∈G} α_{g}g ↦ ∑_{g∈C_{G}(P)} α_{g}g.

It is not difficult to show that σ_{P} is a well defined algebra homomorphism, and its image is even central in F [N_{G}(P)], so that we can view σ_{P} also as a homomorphism Z(F G) → Z(F [N_{G}(P)]).

4.4 Definition Let H ≤ G, let b be a block of F H, and let θ : Z(F H) → F be the central character of F G, so that θ(1_{b}) ≠ 0. If the composition of θ with the natural projection

Z(F G) → Z(F H) → F

∑_{g∈G} α_{g}g ↦ ∑_{g∈H} α_{g}g ↦ θ(∑_{g∈H} α_{g}g)

defines a central character θ^{G} of F G, then we say that b^{G} is defined, and we call the corresponding block B of F G the Brauer correspondent of b, and write B = b^{G}.

if U ≤ H ≤ G, and b is a block of F U, then (b^{H})^{G} = b^{G} provided that all Brauer correspondents are defined.

If b is a block of F H with defect group D, and if b^{G} is defined and has defect group D′ then there exists g ∈ G such that D ≤ ^{g}D′.

Note that even for a fixed subgroup H, b^{G} may be defined for some blocks and not for others. However, under certain conditions on H Brauer correspondents are defined for all blocks.

4.5 Theorem Let P be a p-subgroup of G, and let H be a subgroup of G satisfying P C_{G}(P) ≤ H ≤ N_{G}(P). The the Brauer correspondent B = b^{G} is defined for all blocks b of F H. If θ is the central character of F H corresponding to b, then the central character θ^{G} corresponding to B is given by θ^{G} = θ ∘ σ_{P}.

In the case H = N_{G}(P) one can say even more. This is the first of three main theorems by R. Brauer.

4.6 Theorem (Brauer’s first main theorem) Let P be a p-subgroup of G, and set H = N_{G}(P). Then the Brauer correspondence gives a bijection between the blocks of F H with defect group P and the blocks of F G with defect group P.

Denote by B_{1}(G) the principal block of F G.

4.7 Theorem (Brauer’s third main theorem) Let H ≤ G, let b be a block of F H with defect group P, and assume that P C_{G}(P) ≤ H. Then b^{G} is defined, and one has

b^{G} = B_{1}(G) if and only if b = B_{1}(H).

There is also a connection between the Green correspondence and the Brauer correspondence.

4.8 Theorem Let M be an indecomposable RG-lattice with vertex P, and let N be its Green correspondent, so that N is an indecomposable lattice for H = N_{G}(P). Then M and N belong to blocks B of F G and b of F H respectively, such that B = b^{G}.

Blocks with cyclic defect group are particularly well understood. We just mention one result. To state it, recall that an algebra A is said to be of finite representation type if there is only a finite number of isomorphism classes of indecomposable A-modules. The following theorem would not be true if one replaces F by O.

4.9 Theorem Let B be a block of F G, and let P be its defect group. Then B is of finite representation type if and only if P is cyclic.

We can now state the following refinement of Alperin’s weight conjecture. Let B be a block of F G. Recall that a weight of G is a pair (P, [S]) consisting of a p-subgroup P and an isomorphism class of simple projective F [N_{G}(P)/P]- modules. Given a weight (P, [S]), we can view S as an F N_{G}(H)-module (via inflation), so that S belongs to a certain block b of F N_{G}(P), and b^{G} is defined. We say that the weight (P, [S]) belongs to B if b^{G} = B.

4.10 Conjecture (Alperin’s blockwise weight conjecture) The number of simple F G-modules belonging to a block B of F G equals the number of conjugacy classes of weights belonging to B.

There is an equivalent formulation of this conjecture, due to Knörr and Robinson. To state it, we need some more notation. As before, denote by S_{p}^{∗}(G) be set of non-trivial p-subgroups of G. We will consider chains of subgroups

σ : P_{0} < P_{1} < … P_{n}

with P_{i} ∈ S_{p}^{∗}(G), where we call n = |σ| the length of the chain σ. We also allow the empty chain ∅ and define its length to be |∅| = −1. Denote by Δ_{p}(G) the set of all such chains σ. Then G acts on Δ_{p}(G) by conjugation, i.e., for g ∈ G, the chain ^{g}σ is given by ^{g}P_{0} < ⋯ < ^{g}P_{n}. Denote by N_{G}(σ) the stabilizer of σ in G, then obviously ℕ_{G}(σ) = ⋂_{i=0}^{n} N_{G}(P_{i}).

Knörr and Robinson showed that for every block b of F N_{G}(σ) the Brauer correspondent b^{G} is defined. They also proved that the blockwise weight conjecture can be expressed in the following way. We set

l(H, B) =| {φ ∈ IBr(b) | b a block of F H with b^{G} = B} |

and

k(H, B) =| {χ ∈ Irr(b) | b a block of F H with b^{G} = B} |

for B a block of F G and H a subgroup of G such that the Brauer correspondent of every block of F H is defined.

4.11 Theorem (Knörr-Robinson, 1987) Alperin’s blockwise conjecture 4.10 holds for all primes p, all finite groups G and all blocks B of F G if and only if

∑_{σ∈Δ_{p}(G)/G}(−1)^{|σ|}l(N_{G}(σ), B) = \{ \begin{matrix}0, & if d(B) > 0; \\ −1, & if d(B) = 0 \end{matrix}

for all primes p, all finite groups G and all blocks B of F G.

Moreover, Knörr and Robinson showed that instead of irreducible Brauer characters one can as well count ordinary irreducible characters.

4.12 Theorem One has for every block B of F G that

∑_{σ∈Δ_{p}(G)/G}(−1)^{|σ|}l(N_{G}(σ), B) = ∑_{σ∈Δ_{p}(G)/G}(−1)^{|σ|}k(N_{G}(σ), B)

In 1992, Dade gave a further refinement of these conjectures. We set

k(H, B, d) =| {χ ∈ Irr(b) | b a block of F H with b^{G} = B and d(χ) = d} |

for H a subgroup of G such that the Brauer correspondent b^{G} is defined for all blocks b of F H, B a block of F G and d ≥ 0.

4.13 Conjecture (Dade conjecture) For every block B of F G and every d ≥ 0 one has

∑_{σ∈Δ_{p}(G)/G}(−1)^{|σ|}k(N_{G}(σ), B, d) = \{ \begin{matrix}0, & if d(B) > 0; \\ −1, & if d(B) = 0 \end{matrix}

We also mention another (older) conjecture and its recent refinements. Set

k_{0}(H) =| {χ ∈ Irr(H) | p |̸ χ(1)} |

In other words, k_{0}(H) is the number of characters χ of H with maximal defect, i.e., p^{d(χ)} = |G|_{p}.

4.14 Conjecture (McKay conjecture, 1971) Let P be a Sylow p-subgroup of G. Then

k_{0}(N_{G}(P)) = k_{0}(G).

Of course there is also a blockwise version. For a block B of F H we set

k_{0}(B) =| {χ ∈ Irr(B) | d(χ) = d(B)} | .

4.15 Conjecture (Alperin-McKay, 1975) Let B be a block of F G with defect group P, and let b be the unique block of F N_{G}(P) such that b^{G} = B (given by Brauer’s first main theorem). Then

k_{0}(b) = k_{0}(B).

It was only in 2001 that Isaacs and Navarro observed an even finer structure. We write

M_{k}(H) =| {χ ∈ Irr(H) | χ(1) ≅ ±k mod p} |

for 1 ≤ k ≤ (p − 1)/2. For the primes 2 and 3 the following is of course just the McKay conjecture as stated above.

4.16 Conjecture (Isaacs-Navarro conjecture part A, 2001) Let P be a Sylow p-subgroup of G. Then

M_{k}(N_{G}(P)) = M_{k}(G)

for all integers k ∈ {1, …, (p − 1)/2}.

Now the following should not be too surprising. For a block B of G and an integer k we set

M_{k}(B) =| {χ ∈ Irr(B) | d(χ) = d(B) and χ(1)_{p′} ≅ ±k mod p} | .

4.17 Conjecture (Isaacs-Navarro conjecture part B, 2001) Let B be a block of F G with defect group P, and let b be the block of F N_{G}(P) with b^{G} = B having the same defect group. Then

M_{ck}(B) = M_{k}(b)

for all integers k ∈ {1, …, (p − 1)/2}, where c = |G : N_{G}(P)|_{p′}.

Note that part A follows from part B, since if P is a Sylow p-subgroup of G then c = |G : N_{G}(P)| is congruent to 1 by Sylow’s theorem.

All these conjectures have been shown to be true in many special cases, and it is believed that there must be some deeper structure that is reflected by these numerical relations. However, so far nobody could explain them in the most general case.

In the case of blocks with abelian defect group, a conjecture by M. Broué, and its refinement by J. Rickard offers an explanation for these numerical equalities. They conjecture the existence of what Rickard called a splendid equivalence between the derived module categories of blocks. For details we refer to the articles [5] and [7].

References

[1] Curtis/Reiner Methods of Representation Theory Vol. I. John Wiley & Sons, New York. 1981

[2] Curtis/Reiner Methods of Representation Theory Vol. II. John Wiley & Sons, New York. 1987

[3] J. Alperin Local representation theory. Cambridge University Press, 1986.

[4] J. Alperin: Weights for finite groups. Proc. Sympos. Pure Math. 47 (1987), 369-379

[5] M. Broué: Rickard equivalences and block theory. Groups 93 Galway/St Andrews I LMS lecture notes 211 (1995), 58-79

[6] W. Feit: The representation theory of finite groups. North-Holland Mathematical library, 25 (1982)

[7] J. Rickard: Splendid equivalences: derived categories and permutation modules. Proc. London Math. Soc. (3) 72 (1996), no.2, 331-358

[8] J.Thévenaz G-algebras and modular representation theory Oxford University Press, Oxford 1995