Proceedings of the Summer School on “Representation theory of Algebras, Finite and Reductive Groups” “Babeş-Bolyai” University Cluj, 1998; pp. 1-32.

I. GROUPS, ALGEBRAS, REPRESENTATIONS

Andrei Marcus

The principal aim of this chapter is to introduce the concepts and facts needed in the next chapters. Most of the material is standard and can be found in the books of J.L. Alperin [Al], D. Benson [Be], W. Feit [Fe], and J. Thévenaz [Th]. The survey papers in the reference list also contain extensive bibliography. Our presentation is informal; however we include some proofs (or sketches of proofs) which are both short and instructive. Several proofs are left as exercises, and these should be accessible to readers with a good background in general algebra. At the end we gather enough knowledge to be able to formulate some of the most important conjectures in the field.

1. Introduction

Let F be a field and G a finite group. A matrix representation of degree n ∈ ℕ^{∗} of G over F is a group homomorphism

X : G → GL_{n}(F).

We prefer to consider an F -vector space V with dim_{F} V = n and X : G → Aut_{F} (V). If we denote by F G the F -space having G as a basis, that is

F G = {∑_{g∈G} α_{g}g | α_{g} ∈ F },

then F G is an F -algebra, called the group algebra, with multiplication

(1.1)  (∑_{g∈G} α_{g}g)(∑_{h∈G} β_{h}h) = ∑_{x∈G}( ∑_{gh=x} α_{g}β_{h})x.

It can be easily seen (Exercise!) that giving ρ: G → Aut_{F} (V) is the same as giving an F G-module structure on V by the formula

g · v = ρ(g)(v), ∀g ∈ G, v ∈ V.

Maschke’s theorem tells us that

F G is a semisimple algebra if and only if charF ∤ |G|.

We shall be therefore interested to consider both representations in characteristic 0 and in characteristic p > 0, and also to relate these two kinds of representations. This can be done by considering group algebras over certain commutative local rings. Notice that for any commutative ring R, the group algebra RG is defined in the same manner:

RG = {∑_{g∈G} α_{g}g | α_{g} ∈ R}

is the free R-module with basis G and multiplication defined by (1.1).

2. Complete local rings

2.1. A commutative ring O is local if its radical J(O) is the unique maximal ideal in O, or equivalently, if the quotient k = O/J(O) is a field; in this case, k is called the residue field of O. 2.2. If O is a commutative local Noetherian ring, then the following properties hold.

a) For any finitely generated O−module M we have ∩_{k≥1}J(O)^{k}M = {0}. Indeed, the submodule U = ∩_{k≥1}J(O)^{k}M has the property U = J(O)U and it is finitely generated as M is Noetherian; thus Nakayama’s lemma implies U = {0}. In particular, we have that

∩_{k≥1}J(O)^{k} = {0}.

b) Every finitely generated projective O-module is free (this is another application of Nakayama’s lemma); thus the free module O of rank 1 is up to isomorphism the unique finitely generated projective indecomposable O-module.

c) If A is an O-algebra finitely generated as O-module, then J(O)A ⊆ J(A) and the quotient algebra A/J(O)A becomes a finite-dimensional k -algebra.

An important fact is that A is also complete in the J(A)-adic topology

Moreover, if M is a finitely generated A-module, we have J(A)M = rad(M) ; indeed, the submodules J(A)M and rad(M) are the inverse images in M of the submodules (J(A/J(O)A))(M/J(O)M) and rad(M/J(O)M) of M/J(O)M which coincide. Thus M is annihilated by J(A) if and only if M is a finite direct sum of simple modules.

2.3. A commutative local noetherian ring O is complete if every Cauchy sequence in O converges; that is, if every sequence (λ_{k})_{k≥1} in O satisfying λ_{k+1} − λ_{k} ∈ J(O)^{k} (where k ≥ 1) has a limit λ ∈ O satisfying λ − λ_{k} ∈ J(O)^{k} for all k ≥ 1.

Observe that the limit is unique since the difference λ − λ′ of any two limits λ, λ′ of this sequence lies in ∩_{k≥1}J(O)^{k} = {0}.

If O is complete, then every finitely generated O−module M is complete ; that is, any sequence (m_{k})_{k≥1} in M satisfying m_{k+1} − m_{k} ∈ J(O)^{k}M (where k ≥ 1) has a limit m ∈ M satisfying m − m_{k} ∈ J(O)^{k}M for all k ≥ 1, and again, such a limit is unique.

2.4. The completeness of O can be expressed in a more technical way: observe that the ring O, endowed with the canonical surjections

O ⟶ O/J(O)^{k}, k ≥ 1

is the projective limit lim_{⟵}O/J(O)^{k} of the system of canonical maps

O/J(O)^{k+1} ⟶ O/J(O)^{k}.

If O is not complete, this can be used to define the completion

Ô = lim_{⟵}O/J(O)^{k}.

More explicitely, we can define Ô as the set of equivalence classes of sequences (λ_{k})_{k≥1} in O satisfying λ_{k+1} − λ_{k} ∈ J(O)^{k} for all k ≥ 1, where two such sequences (λ_{k})_{k≥1} and (λ′_{k})_{k≥1} are equivalent if λ_{k} − λ′_{k} ∈ J(O)^{k} for all k ≥ 1. Componentwise sum and multiplication induce a ring structure on Ô; the ring Ô is again local with unique maximal ideal J(Ô) consisting of all equivalence classes of sequences (λ_{k})_{k≥1} in Ô for which λ_{1} ∈ J(O).

The ring Ô is then complete; there is a natural map

O ⟶ Ô, λ ∈ O ↦ \overline{(λ_{k})},

where λ_{k} = λ for all k ≥ 1. This map can be used to characterize Ô in terms of a universal property; moreover, it induces an isomorphism of residue fields k = O/J(O) ≅ Ô/J(Ô).

An important fact is that an O-algebra A is also complete in the J(A)-adic topology (see [Fe, Theorem 9.11]).

Discrete valuation rings an p-modular systems.

2.5. A ring O is called a discrete valuation ring if it is local and a principal ideal domain such that J(O) ≠ {0}. Then J(O) = πO for a prime element π ∈ O, and any non zero proper ideal in J(O) is equal to J(O)^{k} = π^{k}O for some positive integer k. Such a ring comes along with a valuation v : O − {0} ⟶ ℤ mapping a non zero element λ of O to the maximal integer k satisfying λ ∈ J(O)^{k}. Thus λ is invertible in O if and only if v(λ) = 0 and λ ∈ J(O) otherwise.

If O is a discrete valuation ring, then, up to isomorphism, every finitely generated indecomposable O−module is isomorphic to either the free O-module O of rank one or to the module O/J(O)^{n} for some positive integer n. In particular, a finitely generated O-module is projective if and only if it is torsion free.

2.6. Examples. a) Let p be a prime. The subring ℤ_{p} = \{\frac{a}{b}|a, b ∈ ℤ, (b, p) = 1\} of the rational number field is local Noetherian with unique maximal ideal pℤ_{p}. This ring is a discrete valuation ring, but it is not complete; its completion as described above is the ring ℤ̂_{p} of p−adic integers.

b) More generally, if R is a Dedekind ring (for instance, the ring of algebraic integers in a finite extension of ℚ) and π a prime element in R, we obtain a local subring R_{π} = \{\frac{r}{s}|r, s ∈ R, (s, π) = 1\} of the quotient field of R with unique maximal ideal πR_{π}. This is a discrete valuation ring, whose completion R̂_{π} is then a complete discrete valuation ring.

2.7. Definition. Let p be a prime number. We say that the triple (K, O, k) is a pmodular system of O is a complete discrete valuation ring of characteristic 0, K is the field of fractions of O, and the residue field k = O/J(O) has characteristic p.

By the above discussion we see that p-modular systems exist. Moreover, given a positive integer n, there are p-modular systems (K, O, k) such k is algebraically closed and K contains the n-th roots of unity. This is what we shall need in the next sections.

3. Modules

3.1. Let (K, O, k) be a p-modular system and A an O-algebra, free of finite rank as an O-module. We denote kA = k ⊗_{O} A ≃ A/J(O)A and KA = K ⊗_{O} A. We say that (K, O, k) is a splitting p-modular system for A if K is a splitting field for KA and k is a splitting field for kA.

In particular, if A = OG, then KA = KG, kA = kG. Notice that if k is algebraically closed and K contains the |G|-th roots of unity, then (K, O, k) is a splitting p-modular system for OG. In this case we say that K if “big enough” for G and all its subgroups. Actually this condition on K implies that k is a splitting field for kH, for all H ≤ G (see [C-R, Section 17A]).

3.2. By a module we shall always mean a finitely generated left module (unless otherwise stated). An A-module M which is free of finite rank as an O-module is called an A-lattice.

If A = OG, then A-lattices are called integral representations of G, KA-modules are ordinary representations and kA-modules are called modular representations.

3.3. Proposition. If V is a KA-module, then there is an A-lattice M (not unique up to isomorphism) such that K ⊗_{O} M ≃ V .

Proof. Let {v_{1}, …, v_{n}} be a K-basis of V , and let M = Av_{1} + ⋯ + Av_{n} ⊆ V . Then, as an O-module, M is finitely generated torsion free, hence free. Let {x_{1}, …, x_{m}} be an O-basis of M. Then x_{1}, …, x_{m} are linearly independent over K, and the subspace generated by them coincides with V . Consequently, m = n and V ≃ K ⊗_{O} M.

3.4. Characters. Let M be a KA-module, a ∈ KA, and consider the multiplication

μ(a; M): M → M, μ(a; M)(m) = am (∀)m ∈ M.

By definition, the character of KA afforded by M is the map

χ_{M}: KA → K, χ_{M}(a) = trace(μ(a; M))

and define the degree deg χ_{M} by

deg χ_{M} = χ_{M}(1) = dim_{K} M.

Exercises. a) χ_{M} depends only on the isomorphism class of M, and it is K-linear.

b) If 0 → M′ → M → M′′ → 0 is a short exact sequence in KA-mod, then χ_{M} = χ_{M′} + χ_{M′′}.

Assume now that KA is split semisimple and let k = k(KA) be the number of isomorphism classes of simple modules. Let {M_{1}, …, M_{k}} be a system of representatives for these classes, and let {χ_{1}, …, χ_{k}} be the corresponding irreducible characters of KA.

It follows immediately by the Wedderburn Structure Theorem that there is a bijection between {M_{1}, …, M_{k}} and the set {e_{1}, …, e_{k}} of primitive central idempotents of KA, and this set is a K-basis of Z(KA), so

k = k(KA) = dim_{K} Z(KA).

Moreover, we have that

e_{i}KA ≃ M_{deg χ_{i}}(K),

and if ρ is the character of the left regular KA-module _{KA}KA, then

ρ = ∑_{i=1}^{k} χ_{i}(1)χ_{i}.

Finally, a theorem of Frobenius and Schur (see [CR I, Theorem 3.41] implies that χ_{1}, …, χ_{k} are linearly independent over K.

For a subset X of G, denote X̂ = ∑_{x∈X} x ∈ KG and let C_{1}, …, C_{k(G)} be the conjugacy classes of G. With these notations we have:

3.5. Proposition. If (K, O, k) is a splitting p-modular system for OG, then k(KG) = k(G).

Proof. Since KG is semisimple, it follows by Wedderburn Structure Theorem that k(KG) = dim_{K} Z(KG).

Further, observe that Ĉ_{1}, …, Ĉ_{k(G)} form a K-basis of KG; indeed, these elements are linearly independent since the classes are disjoint; if z = ∑_{g∈G} α_{g}g ∈ Z(KG) and h ∈ G, then hzh^{−1} = z, hence α_{hgh^{−1}} = α_{g}.

3.6. In the modular case we have the following theorem of Brauer: the number of simple kG-modules equals the number of conjugacy classes of p′ -elements of G (see [Be, Section 5.3]).

In particular, if P is a p-group, then kP has only one simple module - the trivial module k; consequently kP is a local ring.

The proof in this particular case is easy. Indeed, let M be a simple kP -module, M^{P} = {m ∈ M | gm = m ∀g ∈ P }, 0 ≠ m ∈ M, and let M′ be the additive subgroup of M generated by {gm | g ∈ P }. Then M′ is a finite abelian p-group on which P acts. Since the number of fixed points is congruent to |M′| modulo p, it follows that there is a non-zero fixed element, that is M^{P} is a non-zero kP -submodule of M. Since M is simple, it follows that M^{P} = M and dim_{k} M = 1.

4. Lifting idempotents. Applications

Let O be a complete commutative local noetherian ring with algebraically closed residue field k of characteristic p ≥ 0, and let A be an O-algebra, finitely generated as an Omodule.

Lifting idempotents modulo ideals of A is an important technical tool and we shall use it, together with results on finite dimensional semisimple algebras (assumed to be known), to deduce the basic properties of A-modules. We begin with an exercise which will hopefully serve as a motivation for the interest in idempotents.

Exercise. Let e, f ∈ A be idempotents and M an A-module.

a) Hom_{A}(Ae, M) ≃ eM as eAe-modules.

b) Hom_{A}(Ae, Af) ≃ eAf as (eAe, fAf)-bimodules.

c) End_{A}(Ae)^{op} ≃ eAe as O-algebras.

d) e and f are conjugate if and only if Ae ≃ Af and A(1 − e) ≃ A(1 − f).

4.1. Definition. A conjugacy class of primitive idempotents of A (under the action of U(A)) is called a point of A. We denote by P(A) the set of points of A.

Denote further Ā = A/J(A) and let A → Ā, a ↦ ā be the canonical map.

4.2. Theorem. a) There is a short exact sequence of groups

1 → 1 + J(A) → U(A) → U(Ā) → 1.

b) The canonical map A → Ā induces a bijection P(A) → P(Ā).

c) If e ∈ A is an indempotent and Ĩ is a decomposition (primitive decomposition) of ē in Ā, then there is a decomposition (primitive decomposition) I of e in A such that Ī = Ĩ. Moreover, if I and J are decompositions of e lifting Ĩ, then I and J are conjugate.

Proof. a) If a ∈ A is not invertible, then Aa ≠ A or aA ≠ A. We may therefore assume that there is a maximal left ideal I of A such a ∈ I. Then Ī is a maximal left ideal of Ā and ā ∈ Ī, hence ā is not invertible.

b) Let ẽ ∈ Ā be an idempotent and let a_{1} ∈ A such that ā_{1} = ẽ and let b_{1} = a_{1}^{2} − a_{1} . Define by recurence the sequences

a_{n} = a_{n−1} + b_{n−1} − 2a_{n−1}b_{n−1} and b_{n} = a_{n}^{2} − a_{n}.

One can easily prove by induction that b_{n} ∈ J(A)^{n}, hence lim b_{n} = 0 and (a_{n}) is a Cauchy sequence) in the J(A)-adic topology. If we let e = lim a_{n} then e^{2} = e and ē = ẽ.

Assume that e, f ∈ A are idempotents such that ē, f̄ are conjugate by an element ū ∈ U(Ā), where u ∈ U(A). Replace first f by ufu^{−1} , so we may assume that ē = f̄. Then the element v = 1 − e − f + 2ef belongs to U(A) and e = vfv^{−1} .

Finally, for the statement on primitivity, we may replace A by eAe using that e ∈ A is a primitive idempotent if and only if 0 and e re the only idempotents of eAe (that is, eAe is a local ring), and that J(eAe) = eJ(A)e = eAe ∩ J(A) (so ēĀē ≃ eAe/J(eAe).

c) Replacing again A by eAe, we may assume that e = 1, and let 1 = ī_{1} + ⋯ + ī_{n} in Ā, where Ĩ = {ī_{1}, …, ī_{n}}. By induction, we can lift the decomposition Ĩ \ {i_{n}} to a decomposition f = i_{1} + ⋯ + i_{n−1} of the idempotent f ∈ A. Then if we let i_{n} = 1 − f, we see that i_{n} is an idempotent lifting ī_{n}.

For the last statement, if e = i_{1} + · + i_{n} = i_{1} + · + j_{n}, where ī_{r} = j̄_{r} for 1 ≤ r ≤ n, then w = i_{1}j_{1} + · + i_{n}j_{n} + (1 − e) ∈ (1 + J(A)), since w̄ = 1, and also i_{r}w = i_{r}j_{r} = wj_{r}, hence i_{r} = wj_{r}w^{−1} for 1 ≤ r ≤ n.

This theorem can be used to show that idempotents can be lifted modulo any ideal of A. 4.3. Theorem. Let I be an ideal of A, Ã = A/I and let a ↦ ã the canonical map.

a) The induced map U(A) → U(Ã) is surjective.

b) The map A → Ã induces a bijection between the set P(A − I) of points of A not lying in I and the set P(Ã) of points of Ã.

c) If e ∈ A is an indempotent and Ĩ is a primitive decomposition of ẽ in Ã, then there is a primitive decomposition I of e in A wich lifts Ĩ. Proof. Besides Theorem 4.2, two observations are needed. The first is that the above statements are true if A is a semisimple k -algebra. The second is that since A/J(A) is semisimple, A/J(A) ≃ Ã/J(Ã) × B, where B = (I + J(A))/J(A), and therefore U(A/J(A)) ≃ Ũ(A/J(Ã)) × U(B). It follows that in the diagram

1 ⟶ 1 + J(A) ⟶ U(A) ⟶ U(Ā) ⟶ 1
↓  ↓  ↓
1 ⟶ 1 + J(Ã) ⟶ U(Ã) ⟶ U(Ã) ⟶ 1

the horizontal sequences are exact, while the left hand side and the right hand side maps are surjective; consequently U(A) → U(Ã) is surjective too.

The next applications are also relatively straightforward consequences of Theorem 4.2 and of the properties of finite dimensional semisimple algebras and the reader can fill in the details for himself (see also [Th, Sections 1.3, 1.4 and 1.5]).

4.4. Theorem. There are bijections between P(A), the set Max(A) of maximal ideals of A, the set Irr(A) of simple A-modules and the set Pim(A) of projective indecomposable A-modules, as follows:

• The point α ∈ P(A) corresponds to the maximal ideal 𝔪_{α} of A if and only if e ∉ 𝔪_{α} for some (and then for every) e ∈ α.

• The point α ∈ P(A) corresponds to the simple A-module V (α) if and only if eV (α) ≠ 0 for some (and then for every) e ∈ α.

• The A-module P is projective indecomposable if and only if there is a primitive idempotent e ∈ A such that P ≃ Ae; moreover Ae ≃ Af if and only if e and f are conjugate.

• The indecomposable projective A-module Ae has J(A)e as its unique maximal

submodule, and Ae/J(A)e ≃ V (α), where e ∈ α. Moreover, we have that V (α) ≃ Ae/J(A)e for all e ∈ α, A/𝔪_{α} ≃ End_{A}(V (α)), and A/J(A) ≃ ∏_{α⊂P(A)} A/𝔪_{α}.

Using lifting of idempotents modulo J(O)A ⊆ J(A) we obtain:

4.5. Corollary. Reduction modulo J(O) induces a bijection between the set of projective A-modules and the set of projective kA-modules.

Observing that any two primitive decompositions of the identity element of an Oalgebra are conjugate, we obtain the

4.6. Krull-Schmid Theorem. Let M be an A-module.

a) M decomposes into a direct sum of indecomposable A-modules.

b) If M = ⨁_{i=1}^{n} M_{i} = ⨁_{j=1}^{m} M′_{j} are two such decompositions, then n = m and there is a permutation σ ∈ S_{n} and an A-automorphism ϕ of M such that ϕ(M_{i}) = M′_{σ(i)} for 1 ≤ i ≤ n.

4.7. Corollary. Let e, f ∈ End_{A}(M) be two idempotents. Then the direct summands e(M) and f(M) of M are isomorphic if and only is e and f are conjugate in End_{A}(M).

4.8. Rosenberg’s Lemma. Let e ∈ A be a primitive idempotent and let I be a set of ideals of A. If e ∈ ∑_{I∈I} I, then there is I ∈ I such that e ∈ I.

Indeed, by Theorem 4.2, we may replace A by A/J(A), in which case the statement is trivial.

Notice that the canonical map Ae → Ae/J(A)e is a projective cover, and recall that if M is an A-module, then rad(M) = J(A)M and M/rad(M) is semisimple. From this and the above results is not difficult to deduce the existence of projective covers (see [Th, Theorem 5.10]).

4.9. Theorem. If M is an A-module and f: P → M/rad(M) is a projective cover, then f lifts to a projective cover f̃: P → M.

The Cartan matrix and the decomposition matrix.

4.10. Definition. If α, β ∈ P(A), then, by definition c_{αβ} is the multiplicity of V (α) = Ae/J(A)e as a composition factor of P (β) = Af, where e ∈ α, f ∈ β. The matrix [c_{αβ}]_{α,β∈P(A)} having integer entries is called the Cartan matrix of A.

4.11. Proposition. Assume that A is a k -algebra. Then

c_{αβ} = dim_{k} Hom_{A}(P (α), P (β)) = dim_{k} eAf.

Proof. Notice first that by Schur’s lemma, Hom_{A}(P (α), V (β)) ≃ k if α = β and is 0 otherwise. The equality follows by induction on the composition length of P (β), observing that since P (α) is projective, the short exact sequence

0 → M → P (β) → V (β) → 0

induces the short exact sequence

0 → Hom_{A}(P (α), M) → Hom_{A}(P (α), P (β)) → Hom_{A}(P (α), V (β)) → 0.

4.11. Definition. Let (K, O, k) be a splitting p-modular system for the O−free Oalgebra A.

Let V_{1}, …, V_{t} be the simple KA-modules, and choose the A-lattices M_{1}, …, M_{t} such V_{i} ≃ K ⊗_{O} M_{i}. Let S_{1}, …, S_{s} be the simple kA-modules, and let P_{1}, …, P_{s} be the projective A-modules such that P̄_{j} = k ⊗_{O} P_{j} is the projective cover of S_{j}. Let finally d_{ij} be the multiplicity of S_{j} as a composition factor of k ⊗_{O} M_{i}. The matrix [d_{ij}] is called the decomposition matrix of A and its entries are called decomposition numbers.

4.12. Proposition. Assume that KA is semisimple. Then d_{ij} is equal to the multiplicity of V_{i} as a composition factor of K ⊗_{O} P_{j}.

In particular,

c_{ij} = ∑_{k=1}^{t} d_{ki}d_{kj},

and the decomposition numbers are independent on the choice of M_{i}.

Proof. Indeed,

d_{ij} = dim_{k} Hom_{kA}(P̄_{j}, M̄_{i})

= rank_{O} Hom_{A}(P_{j}, M_{i}) (since P_{j} is projective)

= dim_{K} Hom_{KA}(K ⊗_{O} P_{j}, V_{i})

= multiplicity of V_{i} as a composition factor of K ⊗_{O} P_{j} (since KA is semisimple.)

Central idempotents and blocks.

4.13. The endomorphism algebra of A, regarded as an (A, A)-bimodule (that is as an A ⊗_{O} A^{op} -module) is its center Z(A). It follows that the bimodule summands of A are determined by the central idempotents of A. By the Krull-Schmidt Theorem it follows that there are uniquely determined primitive idempotents e_{1}, …, e_{n} ∈ Z(A), called block idempotents such that 1 = ∑_{i=1}^{n} e_{i}, hence

A = ⨁_{i=1}^{n} e_{i}A.

The algebra B_{i} = e_{i}A is the called a block algebra and has e_{i} as an identity element.

If M is an A-module, then M = ∑_{i=1}^{n} e_{i}M. Therefore, if M is indecomposable, there is a unique e_{i} such M = e_{i}M. We say in this case that M belongs to B_{i}. Sometimes the same term “block” is used to denominate the primitive central idempotent e, the algebra B = eA and the category of B -modules.

Lifting central idempotents is a slightly different matter, and we have the following theorem due to E.C. Dade.

4.14. Theorem. The natural map A → kA = A/J(O)A induces a bijection between the set of central idempotents of A and the set of central idempotents of kA.

Proof. If e is a central idempotent of A then clearly ē is a central idempotent of Ā.

Conversely, let ẽ be a central idempotent of Ā, so there is an idempotent e ∈ A such that ē = ẽ. We need only to show that e is central. We have the Pierce decomposition of A

A = eAe ⊕ eA(1 − e) ⊕ (1 − e)Ae ⊕ (1 − e)A(1 − e).

Since ē ∈ Z(Ā), it follows that ēĀ(1 − ē) = 0. Thus eA(1 − e) ⊆ J(O)A, and hence eA(1 − e) ⊆ eA(1 − e)J(O). By Nakayama’s lemma, eA(1 − e) = 0, and similarly (1 − e)Ae = 0. Consequently,

A = eAe ⊕ eA(1 − e),

so for any x ∈ A there are elements a, b ∈ A such that x = eae + (1 − e)b(1 − e). It follows immediately that e ∈ Z(A).

5. Self-injective and symmetric algebras

Let again O be a complete commutative local noetherian ring with residue field k. 5.1. The duality functor. If A, B are O-algebras and M is an (A, B)-bimodule, then M^{∗} = Hom_{O}(M, O) is a (B, A)-bimodule with multiplication

(bϕa)(m) = ϕ(amb), for a ∈ A, b ∈ B, m ∈ M.

We obtain in this way a functor (−)^{∗}: A-mod-B → B-mod-A which takes surjective maps to injective maps.

Projective and injective lattices.

5.2. Definition. Assume that A is an O-free O-algebra.

An A-module P is called O-projective if whenever f: M → N is an epimorphism of A-modules and g: P → N is a homomorphism such that there is an O-linear map h: P → M with f ∘ h = g, there is an A-linear map h̄: P → M such that f ∘ h̄ = g.

\xymatrix{P \ar@{-->}[d]_{h̄} \ar[dr]^{g} \\ M \ar[r]_{f} & N \ar[r] & 0}

An A-module I is called O-injective if whenever f: N → M is a monomorphism of A-modules and g: N → I is a homomorphism such that there is an O-linear map h: M → I with h ∘ f = g, there is an A-linear map h̄: M → I such that h̄ ∘ f = g.

\xymatrix{0 \ar[r] & N \ar[r]^{f} \ar[dr]_{g} & M \ar@{-->}[d]_{h̄} \\  &  & I}

Exercise. If A is O-free and P is an A-lattice, then the following statements are equivalent:

(i) P is a projective A-module.

(ii) P is an O-projective A-lattice.

(iii) P^{∗} is an O-injective right A-lattice.

5.3. Proposition. Assume that A is O-free and let M be an A-lattice. Then M has an O-injective hull, unique up to isomorphism.

Proof. We have seen in the preceding section that the right A-lattice M^{∗} has a projective cover f: P → M^{∗}. Since M^{∗} is O-free, there is an O-linear section s: M^{∗} → P of f. Then, since M^{∗∗} ≃ M, we have that f^{∗}: P^{∗} → M is and O-injective hull of M with O-linear retraction s^{∗}.

Self-injective algebras.

5.4. Definition. The O-free O-algebra A is called self-injective if _{A}A is an O-injective A-lattice.

More generally, the (not necessarily O-free) O-algebra A is called O-self-injective if kA is an injective kA-module.

5.5. Proposition. Assume that A is self-injective and let M be an A-lattice. Then M is projective if and only if M is O-injective.

Proof. Since _{A}A is O-injective, it follows that any projective lattice is O injective. By the above exercise, there is a bijection between the set of (isomorphism classes) of O-injective lattices and the set of projective right lattices.

We also have bijections between the set P(A) of points of A and the set of projective left lattices on the one hand, and between P(A) and the set of projective right lattices on the other hand.

Since all these sets are finite, we conclude that the set of projective (left) lattices coincides with the set of O-injective lattices.

Symmetric algebras.

These are important examples of self-injective algebras, and are defined by condition easier to verify.

Recall that if M and N are O-modules and β: M × N → O is an O-bilinear map, then one can define the O-linear maps

θ: M → N^{∗}, θ(x)(y) = β(x, y)

and

θ′: N → M^{∗}, θ(y)(x) = β(x, y).

Then β is called non-degenerate if θ, θ′ are injective, and unimodular if θ, θ′ are surjective; if O is a field, then these two conditions coincide.

5.6. Proposition. The following conditions on the O-algebra A are equivalent:

(i) There is an isomorphism θ: A_{A} → (_{A}A)^{∗} of right A-modules such that θ(a)(b) = θ(b)(a) for a, b ∈ A (or equivalently, θ is an isomorphism of (A, A)-bimodules).

(ii) There is a unimodular form β: A × A → O such that β is symmetric (β(a, b) = β(b, a) ) and associative (β(ab, c) = β(a, bc) for a, b, c ∈ A).

(iii) There is an O-linear map σ: A → O satisfying • σ is symmetric, that is, σ(ab) = σ(ba) for a, b ∈ A.

• Ker(σ) contains no non-zero right ideal of A.

• If f: A → O is an O-linear map, then there is a ∈ A such f(b) = σ(ab) for all b ∈ B.

The proof of the proposition is routine and is left as an exercise. Notice only that the link between θ, β and σ is given by

β(a, b) = σ(ab) = θ(a)(b).

If the above conditions are satisfied, then A is called a symmetric algebra and σ is a symmetrizing form of A. Condition (i) implies immediately that A is self-injective.

5.7. Examples. a) OG is a symmetric algebra with symmetrizing form

σ: OG → O, σ(∑ a_{g}g) = a_{1}.

(If I is a right ideal of A, ∑ a_{g}g ∈ I and a_{h} ≠ 0, then (∑ a_{g}g)h^{−1} ∈ I and (∑ a_{g}g)h^{−1} ∉ Ker σ.)

b) If (A, σ) is a symmetric O-algebra, then M_{n}(O) is symmetric with symmetrizing form σ̄ defined by σ̄(X) = σ(trace(X)) for any X ∈ M_{n}(O).

c) If F is a field and A is a semisimple F -algebra then A is symmetric. Indeed, it suffices to take A = M_{n}(D), where D is a division F -algebra, and use that the reduced trace Trd: D → F of D is a symmetric form.

d) If (A, σ) is a symmetric O-algebra and e ∈ A is an idempotent, then eAe is symmetric by taking the restriction of σ to eAe.

e) If A, B is a symmetric O-algebras, then A × B and A ⊗_{O} B are symmetric too.

5.8. Proposition. Assume that (A, σ) is a symmetric k -algebra and let P be an indecomposable projective A-module, then P/rad(P) ≃ Soc(P).

Proof. If P = Ae, where e is a primitive idempotent of A, then Soc(P) = Soc(P)e is a left ideal if A, so there is x ∈ Soc(P) such that σ(xe) ≠ 0. By symmetry, σ(ex) ≠ 0, hence e Soc(P) ≠ 0. But e Soc(P) ≃ Hom_{A}(P, Soc(P)), so there is a nonzero map P → Soc(P). Since Soc(P) is simple, it follows that P/rad(P) ≃ Soc(P).

5.9. Next we present another useful property which is in connection with Morita equivalences between symmetric algebras.

Let A, B be O−algebras such that (A, σ, β) is symmetric, and let M be an (A, B)- bimodule.

Then there is a natural isomorphism of (B, A)-bimodules

Ψ_{M}: Hom_{A}(id_{M}, σ): Hom_{A}(M, A) ≅ M^{∗}

mapping φ ∈ Hom_{A}(M, A) to σ ∘ φ ∈ M^{∗}.

To construct the inverse of Ψ_{M} we need the so called dual bases. Let {a_{1}, …, a_{n}} be an O-basis of A, and let {b_{1}, …, b_{n}} the dual basis with respect to β, that is, β(a_{i}, b_{j}) = δ_{ij}. Then for any x ∈ A we have

x = ∑_{i=1}^{n} β(x, b_{i})a_{i} = ∑_{j=1}^{n} β(a_{j}, x)b_{j}.

Then the inverse of Ψ_{M} is given by m^{∗} ↦ φ_{m^{∗}}, where φ_{m^{∗}}(m) = ∑_{i=1}^{n} m^{∗}(b_{i}m)a_{i}, for m^{∗} ∈ M^{∗} and m ∈ M.

Exercise. Show that the definition of Ψ_{M}^{−1} does not depend on the choice of dual bases, and that Ψ_{M}^{−1} is indeed the inverse of Ψ_{M}.

5.10. Orthogonality relations. At the end of this section we return to characters. Assume that KA is a symmetric split semisimple K-algebra, where charK = 0, let {a_{1}, …, a_{n}}, {b_{1}, …, b_{n}} be dual bases of KA with respect to the symmetric associative bilinear form β: KA × KA → K and denote z = ∑_{i=1}^{n} a_{i}b_{i}.

As in 3.4, let χ_{1}, χ_{k} be the irreducible characters of KA and e_{1}, …, e_{k} ∈ Z(KA) the corresponding primitive central idempotents. For i = 1, …, k, define d_{i} = ∑_{i=1}^{n} χ_{i}(a)b_{i}; then we clearly have that χ_{i}(a) = β(d_{i}, a) for all a ∈ KA. The next result is referred as the orthogonality relations for characters (see [CR I, Propositions 9.17 and 9.19]).

5.11. Theorem. With the above notations, the following statements hold for 1 ≤ i, j ≤ k :

a) χ_{i}(d_{i}) ≠ 0 and e_{i} = χ_{i}(1)χ_{i}(d_{i})^{−1}d_{i}.

b) χ_{i}(d_{i}) = χ_{i}(z)χ_{i}(1)^{−1}.

c) ∑_{r=1}^{n} χ_{i}(a_{r})χ_{j}(b_{r}) = \{ \begin{matrix}χ(z)χ(1)^{−1}, & if i = j \\ 0 & if i ≠ j. \end{matrix}

6. Operations with kG-modules

In this section let k be any commutative ring, G a finite group, H a subgroup of G.

6.1. A first set of operations arise from the fact that A is not only a k -algebra, but a (cocomutative) Hopf k -algebra (Exercise!) with comultiplication

Δ: A → A ⊗_{k} A, g ↦ g ⊗ g,

counit (or augmentation)

ϵ: A → k, g ↦ 1,

and antipode

S: A → A, g ↦ g^{−1}.

Let M and N be kG-modules. Using that Δ and ϵ are k -algebra maps and S is an antiisomorphism, we have the following constructions:

a) k has a structure of a trivial module via ϵ, sometimes denoted by 1_{G}.

b) (tensor product) M ⊗_{k} N is an kG ⊗_{k} kG-module, hence a kG-module via Δ, with the diagonal action

g(m ⊗ n) = gm ⊗ gn.

c) M becomes a right kG-module via S, so by definition

mg = g^{−1}m.

d) The k -dual M^{∗} = Hom_{k}(M, k) of M is a kG-module with

gϕ(m) = ϕ(g^{−1}m).

e) More generally, Hom_{k}(M, N) is a kG-module with

gϕ(m) = gϕ(g^{−1}m).

Exercise. Show that there is a natural isomorphism of kG-modules

Hom_{k}(M, N) ≃ M^{∗} ⊗_{k} N.

6.2. Restriction, induction, coinduction. Another set of constructions are consequences of the fact that kG is a free (left and right) kH -module of finite rank. Indeed, denote by [G/H] (respectively [H \ G]) a complete set of representatives, containing 1 ∈ G, for the left (respectively right) cosets of H in G. Then

kG = ⨁_{g∈[G/H]} gkH = ⨁_{g∈[H\G]} kHg.

We have then the following exact functors:

• Res_{H}^{G}: kG-mod → kH-mod, Res_{H}^{G} M = M regarded as a kH -module by scalar restriction;

• Ind_{H}^{G}: kH-mod → kH-mod, Ind_{H}^{G} N = kG ⊗_{kH} N;

• Coind_{H}^{G}: kH-mod → kH-mod, Coind_{H}^{G} N = Hom_{kH}(kG, N), where (gϕ)(g′) =

ϕ(g′g); We also have, for any g ∈ G, an equivalence of categories, called conjugation, between

kH-mod and k^{g}H-mod, given by

N ↦ ^{g}N = kg ⊗_{k} M,

where, by definition, (ghg^{−1})(g ⊗ m) = g ⊗ hm.

6.3. Nakayama relations, Eckmann-Shapiro Lemma. Res_{H}^{G} has Ind_{H}^{G} as a left adjoint and Coind_{H}^{G} as a right adjoint, that is, for M ∈ kG-mod and N ∈ kH-mod, there are natural k -linear isomorphisms

• Hom_{kH}(N, Res_{H}^{G} M) ≃ Hom_{kG}(Ind_{H}^{G} N, M);

• Hom_{kH}(Res_{H}^{G} M, N) ≃ Hom_{kG}(M, Coind_{H}^{G} N). Moreover, there is a natural isomorphism Φ: Ind_{H}^{G} N → Coind_{H}^{G} N, given by

Φ(g ⊗ n)(g′) = g′gn, if g′g ∈ H and 0 otherwise

with inverse

Φ^{−1}(ϕ) = ∑_{g∈[G/H]} g ⊗ ϕ(g^{−1}).

Exercise. a) Prove the existence of the above natural isomorphisms. b) Give explicitely the isomorphism Hom_{kH}(Res_{H}^{G} M, N) ≃ Hom_{kG}(M, Ind_{H}^{G} N).

c) Find the units and the counits of the adjunctions. 6.4. Frobenius reciprocity. (Exercise.) With M and N as above, there is a natural isomorphism of kG-modules

Ind_{H}^{G} N ⊗_{k} M ≃ Ind_{H}^{G}(N ⊗_{k} Res_{H}^{G} M),

given by (g ⊗ n) ⊗ m ↦ g ⊗ (n ⊗ g^{−1}m), with inverse g ⊗ (n ⊗ m) ↦ (g ⊗ n) ⊗ gm), for g ∈ G, m ∈ M and n ∈ N. Using exercise 7.1, deduce the isomorphism

Ind_{H}^{G}(Hom_{O}(N, Res_{H}^{G} M) ≃ Hom_{O}(Ind_{H}^{G} N, M).

Mackey Formulas. 6.5. These formulas involve the bimodule structure of kG. Let H and K be subgroups of G and let M be an kK -module. Our aim is to decompose Res_{H}^{G} Ind_{K}^{G} M.

Let [H \G/K] be a system of representatives for the double coset of (H, K) in G, that is, G is the disjoint union of the subsets HgK, for g ∈ [H \ G/K]. Then we have

a) _{kK}kG_{kK} = ⨁_{g∈[H\G/K]} k(HgK).

b) There is an isomorphism of (kH, kK)-bimodules

k(HgK) ≃ kH ⊗_{k(H∩^{g}K)} ^{g}kK

sending hgk to h ⊗ (g ⊗ k). 6.6. Mackey Decomposition Theorem. If M is an kK -module, then there is a natural isomorphism of kH -modules

Res_{H}^{G} Ind_{H}^{G} M ≃ ⨁_{g∈[H\G/K]} Ind_{H∩^{g}K}^{H} Res_{H∩^{g}K}^{^{g}K} ^{g}M.

Proof. Using the above remarks a) and b), we have

Res_{H}^{G} Ind_{K}^{G} M ≃ _{kH}kG_{kK} ⊗_{kK} M

≃ ⨁_{g∈[H\G/K]} k(HgK) ⊗_{kK} M

≃ ⨁_{g∈[H\G/K]}(kH ⊗_{k(H∩^{g}K)} ^{g}kK) ⊗_{kK} M

≃ ⨁_{g∈[H\G/K]} kH ⊗_{k(H∩^{g}K)} Res_{H∩^{g}K}^{^{g}K}(g ⊗ M)

≃ ⨁_{g∈[H\G/K]} Ind_{H∩^{g}K}^{H} Res_{H∩^{g}K}^{^{g}K} ^{g}M.

Exercise. Let M be an kK -module and N a kH -module.

a) Ind_{K}^{G} M ⊗_{O} Ind_{H}^{G} N ≃ ⨁_{g∈[H\G/K]} Ind_{H∩^{g}K}^{G}(Res_{H∩^{g}K}^{^{g}K} ^{g}M ⊗_{O} Res_{H∩^{g}K}^{^{g}K} N).

b) Hom_{k}(Ind_{K}^{G} M, Ind_{H}^{G} N) ≃ ⨁_{g∈[H\G/K]} Ind_{H∩^{g}K}^{G} Hom_{k}(Res_{H∩^{g}K}^{^{g}K} ^{g}M, Res_{H∩^{g}K}^{^{g}K} N).

c) Ind_{K}^{G} M ⊗_{kG} Ind_{H}^{G} N ≃ ⨁_{g∈[H\G/K]}(Res_{H∩^{g}K}^{^{g}K} ^{g}M ⊗_{k(H∩^{g}K)} Res_{H∩^{g}K}^{^{g}K} N).

d) Hom_{kG}(Ind_{K}^{G} M, Ind_{H}^{G} N) ≃ ⨁_{g∈[H\G/K]} Hom_{k(H∩^{g}K)}(Res_{H∩^{g}K}^{^{g}K} ^{g}M, Res_{H∩^{g}K}^{^{g}K} N).

7. Relative projectivity

We continue to use the general assumption of the preceding section.

7.1. Definition. a) The kG-module M is called relatively H -projective if whenever f: N → N′ is an epimorphism of kG-modules and g: M → N′ is a homomorphism such that there is a kH -linear map h: M → N with f ∘ h = g, there is an kG-linear map h̄: M → N such that f ∘ h̄ = g.

\xymatrix{M \ar@{-->}[d]_{h̄} \ar[dr]^{g} \\ N \ar[r]_{f} & N′ \ar[r] & 0}

b) The kG-module M is called relatively H -injective if whenever f: N′ → N is a monomorphism of kG-modules and g: N′ → M is a homomorphism such that there is a kH -linear map h: N → M with h ∘ f = g, there is an kG-linear map h̄: N → M such that h̄ ∘ f = g.

\xymatrix{0 \ar[r] & N′ \ar[r]^{f} \ar[dr]_{g} & N \ar@{-->}[d]_{h̄} \\  &  & M}

c) The short exact sequence 0 → N′ → N → N” → 0 of kG-modules is called H -split, if it splits as a sequence of kH -modules.

7.2. The trace map. Let M, N be kG-modules. We have seen that G acts on Hom_{k}(M, N), and it is easy to show that the set Hom_{k}(M, N)^{H} of elements fixed by H coincides with Hom_{kH}(M, N). Then the trace map is, by definition, the map

Tr_{H}^{G}: Hom_{kH}(M, N) → Hom_{kG}(M, N)

given by

Tr_{H}^{G}(α)(m) = ∑_{g∈[G/H]}(^{g}α)(m).

Exercise. a) Show that Tr_{H}^{G}(α) is independent on the choice of [G/H] and that it is indeed a kG-linear map.

b) Let α: M_{1} → M_{2} and β: M_{2} → M_{3} . If α is kH -linear and β is kG-linear, then β ∘ Tr_{H}^{G}(α) = Tr_{H}^{G}(β ∘ α); if α is kG-linear and β is kH -linear, then Tr_{H}^{G}(β) ∘ α = Tr_{H}^{G}(β ∘ α).

It follows that Im Tr_{H}^{G} is an “ideal” of Hom_{kG}(M, N).

c) (Transitivity.) If H ≤ K ≤ G, then Tr_{K}^{G} ∘ Tr_{H}^{K} = Tr_{H}^{G}.

The trace map can be used to characterize relative projectivity.

7.3. Higman’s Criterion. If M is a kG-module and H ≤ G, then the following statements are equivalent:

(i) M is relatively H -projective.

(ii) If N →^{α} M → 0 is an H -split map, then α splits.

(iii) M is relatively H -injective.

(iv) If 0 → M →^{α} N is an H -split map, then α splits.

(v) M is a direct summand of Ind_{H}^{G} M ≃ Coind_{H}^{G} M.

(vi) There is a kH -module N such that M is a direct summand of Ind_{H}^{G} N.

(vii) id_{M} belongs to Tr_{H}^{G}(End_{kH}(M).

Proof. Observe that

Ind_{H}^{G} M →^{μ} M → 0, g ⊗ m ↦ gm

is a kG-epimorphism with kH -linear section

M → Ind_{H}^{G} M, m ↦ 1 ⊗ m.

From this, one obtains the implications (i) ⇒ (ii) ⇒ (v) ⇒ (vi).

Similarly, we have that

0 → M → Coind_{H}^{G} M = Hom_{kH}(kG, M), m ↦ f_{m}, f_{m}(1) = m

is a kG-linear monomorphism with kH -linear retraction

Coind_{H}^{G} M → M, f ↦ f(1).

From this, one obtains the implications (iii) ⇒ (iv) ⇒ (v).

(vi)⇒(vii). Assume that M is a direct summand of Ind_{H}^{G} N and let θ be the composite

M ↪ Ind_{H}^{G} N →^{π} Ind_{H}^{G} N ↠ M,

where π(g ⊗ n) = g ⊗ n if g ∈ G and 0 otherwise. Then Tr_{H}^{G} π = id_{Ind_{H}^{G} N} and Tr_{H}^{G}(θ) = id_{M}.

With the notations of Definition 7.1.a), let θ ∈ End_{kH}(M) such that Tr_{H}^{G}(θ) = id_{M}, and let h̄ = Tr_{H}^{G}(h ∘ θ). Then

f ∘ h̄ = f ∘ Tr_{H}^{G}(h ∘ θ) = Tr_{H}^{G}(f ∘ h ∘ θ)
= Tr_{H}^{G}(g ∘ θ) = g ∘ Tr_{H}^{G}(θ) = g.

A dual proof works for (vii)⇒(iii).

We also have a notion of relative projectivity for maps.

7.4. Proposition. Let α ∈ Hom_{kG}(M_{1}, M_{2}). The following statements are equivalent: (i) α ∈ Tr_{H}^{G}(Hom_{kH}(M_{1}, M_{2})). (ii) α is the composition of the obvious maps M_{1} → Ind_{H}^{G} Res_{H}^{G} M_{2} →^{μ} M_{2} . (iii) α factors through a relatively H -projective kG-module M.

Proof. (i)⇒ (ii). α = Tr_{H}^{G}(θ) = μ∘Ψ(θ), where Ψ is the isomorphism Hom_{kH}(M_{1}, M_{2}) ≃ Hom_{kG}(M_{1}, Ind_{H}^{G} M_{2}).

(iii)⇒(i). If id_{M} = Tr_{H}^{G}(θ) and α = α_{2} ∘ α_{1}, then

α = α_{2} ∘ Tr_{H}^{G} ∘α_{1} = Tr_{H}^{G}(α_{2} ∘ θ ∘ α_{1}) ∈ Im Tr_{H}^{G} .

If the above conditions are satisfied, then α is called a relatively H -projective map. Using Frobenius reciprocity, we deduce

7.5. Corollary. Let M and N be kG-modules. If M or N is relatively H -projective, then M ⊗_{k} N is relatively H -prohective.

7.6. Corollary. If [G : H] is invertible in k, then every kG-module M is relatively H -projective.

Proof. Observe that Tr_{H}^{G}([G : H]^{−1} · id_{M}) = id_{M}.

7.7. Maschke’s Theorem. Let F be a field. Then F G is semisimple if and only if the characteristic of F does not divide the order of G.

Proof. One implication is immediate by Corollary 8.6. Conversely, assume that charF | |G|, and let G̃ = ∑_{g∈G} g ∈ F G. Observe that for any x ∈ G, xG̃ = G̃x = G̃, so, in general, if a ∈ F G, then aG̃ = ϵ(a)G̃. It follows that F G̃ is a one dimensional ideal of F G. Since G̃^{2} = |G| · G̃ = 0, we have that F G̃ is a nipotent ideal, hence J(F G) ≠ 0.

7.8. Remark. Actually, if the characteristic of F does not divide the order of G, then F G is more than semisimple, it is separable, since the multiplication

μ: F G ⊗_{F} F G → F G, g ⊗ h ↦ gh

is split as a bimodule map by g ↦ |G|^{−1} ∑_{g∈G} g ⊗ g^{−1}.

8. Vertices and sources. The defect group of a block

We return to the complete local commutative ring O with residue field k of characteristic p > 0 dividing the order of G. Assume that the p′ -part of |G| is invertible in O, and let M be an indecomposable OG-module. We are interested to find the minimal subgroups H of G such that M is relatively H -projective.

8.1. Definition. A subgroup P of G is called a vertex of M if it is minimal with respect to the property that M is relatively P -projective.

If P is a vertex of M and U is an indecomposable O-module such that M is a direct summand of Ind_{P}^{G} U, then U is called a source of M.

8.2. Proposition(J.A. Green). a) The vertices of M form a G-conjugacy class of p-subgroups of G.

b) If the OP -modules U and V are vertices of M, then there is g ∈ N_{G}(P) such that V ≃ ^{g}U.

Proof. The fact that the vertices are p-subgroups is clear by Higman’s Criterion.

If H, K ≤ G and M is a direct summand of both Ind_{H}^{G} Res_{H}^{G} M and Ind_{K}^{G} Res_{K}^{G} M then, by Mackey’s formula M is a summand of

Ind_{H}^{G} Res_{H}^{G} Ind_{K}^{G} Res_{K}^{G} M ≃ ⨁_{g∈[H\G/K]} Ind_{H∩^{g}K}^{G} Res_{H∩^{g}K}^{G} M.

Consequently, if H and K are both vertices, then H = ^{g}K for some g ∈ G.

b) By Higman’s Criterion, there is a summand W of Res_{P}^{G} M such that W is a source of M. Then W is a summand of

Res_{P}^{G} Ind_{P}^{G} U ≃ ⨁_{g∈[P \G/P]} Ind_{P ∩^{g}P}^{P} Res_{P ∩^{g}P}^{P} ^{g}U,

hence by Krull-Schmidt Theorem, W ≃ ^{g}U for some g ∈ N_{G}(P).

8.3. The defect group of a block. We pursue a module theoretic point of view on blocks. Observe first, that (OG)^{op} is naturally isomorphic to O via the augmentation S, and that OG ⊗_{O} OG ≃ O(G × G) by g ⊗ h ↦ (g, h). It follows that OG is an O(G × G)-module with the multiplication (g, h)a = gah^{−1} , and by remark 7.5.b),

OG ≃ O(G × G) ⊗_{δ(G)} O,

where δ: G → GG is the diagonal map.

A block B of OG is an indecomposable O(G × G)-summand of OG, hence it is relatively δ(G)-projective. The subgroup D of G is called a defect group of B if δ(D) is a vertex of the O(G × G)-module B. (This is one of the many definitions of the defect group.)

8.4. Remarks. a) The defect groups of B form a conjugacy class of p-subgroups of G. If the order of a defect group of B is p^{d}, then d is called the defect of B.

b) If B = eOG has defect group D and M is an indecomposable OG-module belonging to B, then M is relatively D-projective.

Indeed, regard B as a OG-module by scalar restriction via δ. The map M → B ⊗_{O} M, m ↦ e ⊗ m is OG-linear and has a retraction induced by multiplication, hence M is a direct summand of B ⊗_{O} M. It is not difficult to see that Res_{G}^{G×G} B is still relatively D-projective, so, by Frobenius reciprocity, B ⊗_{O} M, and hence M are relatively Dprojective.

c) The block containing the trivial OG-module O is the principal block of OG. Since the trivial module has as vertices the Sylow p-subgroups of G, it follows that these are also the defect groups of the principal block of OG.

d) The bijection between the blocks of OG and the blocks of kG induced by reduction modulo J(O) preserves defect groups.

8.5. Brauer’s conjecture on Cartan invariants and decomposition numbers. For every integer d, the Cartan invariants and the decomposition numbers of a block of defect d are bounded in terms of d.

8.6. Feit’s conjecture. For every p-group P, there exist only finitely many isomorphism classes of indecomposable OP -modules which are sources of simple OG-modules, where G contains P.

8.7. Donovan’s conjecture. For every p-group D, there exist only finitely many Morita equivalence classes of blocks with defect group D.

9. The Green correspondence and the Brauer correspondence

As in the preceding section, assume that O is a complete commutative local noetherian ring with residue field k of characteristic p > 0 dividing |G| such that the p′ -part of |G| is invertible in O.

Let P be a p-subgroup of G and let H = N_{G}(P). Denote

X = {X ≤ G | X ≤ ^{g}P ∩ P for some g ∈ G \ H},
Y = {Y ≤ G | Y ≤ ^{g}P ∩ H for some g ∈ G \ H},

and observe X ⊆ Y and P ∉ Y. The following theorem is a classical version of Green’s correspondence, which reduces certain problems about OG-modules to problems about OH -modules.

9.1. Theorem. There is a bijective correspondence between the indecomposable OGmodules with vertex P and the indecomposable OH -modules with vertex P defined as follows.

a) If U is an indecomposable OG-module with vertex P, then Res_{H}^{G} U has a unique indecomposable summand f(U) with vertex P, and the remaining summands have vertices in Y.

b) If V is an indecomposable OH -module with vertex P, then Ind_{H}^{G} V has a unique indecomposable summand g(V) with vertex P, and the remaining summands have vertices in X.

c) g(f(U)) ≃ U and f(g(V)) ≃ V .

Proof. The proof is a careful application of Mackey decomposition and of the Krull-Schmidt theorem.

a) If W is a source of U, then there is an indecomposable OH -module V such that U is a summand of Ind_{H}^{G} V , and let Ind_{P}^{H} W ≃ V ⊕ V′. We have

Res_{H}^{G} Ind_{P}^{G} W ≃ Res_{H}^{G} Ind_{H}^{G} Ind_{P}^{H} W
≃ Res_{H}^{G} Ind_{H}^{G} V ⊕ Res_{H}^{G} Ind_{H}^{G} V′.

By Mackey decomposition we also have that

Res_{H}^{G} Ind_{P}^{G} W ≃ Ind_{P}^{H} W ⊕ W′,

where is an OH -module whose indecomposable summands are projective relative to subgroups Y ∈ Y. It follows that

Res_{H}^{G} Ind_{H}^{G} V ⊕ Res_{H}^{G} Ind_{H}^{G} V′ ≃ V ⊕ V′ ⊕ W′,

so by the Krull Schmidt theorem,

Res_{H}^{G} Ind_{H}^{G} V ≃ V ⊕ V′′,

where the indecomposable summands of V′′ are projective relative to subgroups Y ∈ Y. Since U is a summand of Ind_{H}^{G} Res_{H}^{G} U, it follows that some summand of Res_{H}^{G} U has vertex P, so ve define f(U) = V .

b) There is an indecomposable summand U of Ind_{H}^{G} V such that V is a summand of Res_{H}^{G} U, and let

Ind_{H}^{G} V = U ⊕ U′.

Then U′ is relatively P -projective, and assume that there is an indecomposable summand U_{1} of U′ which is not projective relative to a subgroup X ∈ X. Let P_{1} ≤ P be a vertex and W_{1} ∈ OP -mod be a source of U_{1} . Then W_{1} is a summand of Res_{P_{1}}^{G} U_{1} , so there is indecomposable summand V_{1} of Res_{H}^{G} U_{1} such that V_{1} is a summand of Res_{P_{1}}^{H} V_{1} . It follows that V_{1} , and therefore Res_{H}^{G} U′ are not projective relative to subgroups Y ∈ Y, which is a contradiction.

Thus the summands of U′ are projective relative to subgroups in X, and we define g(V) = U.

c) is a consequence of a) and b).

9.2. Remarks. a) Since the Green correspondence is induced by induction and restriction, it is not surprizing that it actually gives an equivalence between certain quotient categories. Recently, M. Auslander and M. Kleiner showed that there is a very general categorical setting for the Green correspondence and we refer to [Z] for details and applications.

b) There is also the following strengthening of the Green correspondence due to D. Burry, J. Carlson and L. Puig (see [B, Theorem 3.12.3]).

If U is an indecomposable OG-module and V is an indecomposable summand of Res_{H}^{G} U with vertex P, then U=g(V).

9.3. The Brauer correspondence. Let B be a block of OG (that is, an indecomposable O(C)-summand of OG) with defect group D and let H = N_{G}(D). Observe that N_{G×G}(D × D) = N_{G}(D) × N_{G}(D), and as an O(H × H)-module,

OG ≃ OH ⊕ ⨁_{g∈[H\G/H],g∉H} O(HgH).

Then B has a Green correspondent b with respect to G × G, H × H, D × D, which turns out to be a block of OH with defect D.

One obtains in this way a bijection between the blocks of OG with defect group D and blocks of OH with defect group D, called the Brauer correspondence (this is one of its many definitions).

9.4. Remarks. a) The Brauer correspondence is compatible with the Green correspondence (see [Al, Section 14]) and sends the principal block of OG to the principal block of OH.

An interesting case when this is useful is the so called trivial intersection (TI) situation, that is, when D ∩^{g}D = {1} for all g ∈ G \ N_{G}(D). Then the induction and restriction induce a stable equivalence between B and b; see Chapter III, section 6.

b) Brauer correspondence can be defined in a more general situation. Let D be a p-subgroup of G, H a subgroup containing the centralizer C_{G}(D) and let b be a block of OH. Then there is a unique block B = b^{G} of OG such that such b is a summand of Res_{H×H}^{G×G}. We also say that b is a Brauer correspondent of B.

10. Clifford theory

Let O and G be as in the preceding section, and for simplicity, we shall assume that k is algebraically closed.

We have seen that the Green correspondent of an indecomposable OG-module is projective relative to a normal subgroup. The systematic study of the relation between the representations of a group and the representations of its normal subgroup was initiated by A.H. Clifford, and his theory can be used to give inductive arguments which allow to reduce certain problems to the case of simple groups. We also mention that Clifford theory is particularly useful in the representation theory of p-solvable groups. Recall that G is p-solvable if it has a chain

1 = N_{1} < N_{2} < ⋯ < N_{r} = G

of normal subgroups such that N_{i}/N_{i−1} is a p-group or a p′ -group. Our presentation in this section is in the spirit of E.C. Dade, who introduced the language of group-graded algebras.

Let H be a finite group, N a normal subgroup such that G = H/N, and denote A = OH. Then A has a decomposition A = ⨁_{g∈G} A_{g} into O-submodules, where A_{g} = ONh for h ∈ g, such that A_{g}A_{g′} = A_{gg′} for all g, g′ ∈ G. It will be convenient to introduce the following definition.

10.1. Definition. a) An O-algebra A is called G-graded if

A = ⨁_{g∈G} A_{g} as O-modules

and A_{g}A_{h} ⊆ A_{gh} for all g, h ∈ G.

b) A is strongly graded if, in addition,

A_{g}A_{h} = A_{gh} for all g, h ∈ G.

c) An A-module M is called G-graded if

M = ⨁_{x∈G} M_{x} as O-modules

and A_{g}M_{x} ⊆ M_{gx} for all g, x ∈ G.

For a subset X ⊆ G we denote M_{X} = ⨁_{x∈X} M_{x}.

d) An A-linear map f: M → M′ between two G-graded A-modules is graded if f(M_{x}) ⊆ M′_{x} for all x ∈ G.

We denote by A-gr the category of G-graded A-modules and graded A-maps.

e) A submodule N of M is graded if for every n = ∑_{x∈G} n_{x} ∈ N and for every x ∈ G, the homogenous component n_{x} belongs to N.

10.2 Dade’s Theorem. If A is a G-graded O-algebra, then the following statements are equivalent:

(i) A is strongly graded.

(ii) The induction functor

A ⊗_{A_{1}} −: A_{1}-mod → A-gr

and the truncation functor

(−)_{1} : A-gr → A_{1}-mod

are inverse equivalence of categories.

(iii) For each g ∈ G, A_{g} ⊗_{A_{1}} −: A_{1}-mod → A_{1}-mod is an equivalence with inverse A_{g^{−1}} ⊗_{A_{1}} −.

10.3. Crossed products and twisted group algebras. The set

hU(A) = ⋃_{g∈G}(U(A) ∩ A_{g})

of homogenous units of A is a subgroup of U(A), and deg: hU(A) → G is a grouphomomorphism with kernel U(A_{1}). This map is surjective if A has a homogenous invertible element in each degree, and in this case, A is called a crossed product of A_{1} and G. (Notice that if N ⊴ H and G = H/N, then OH is a crossed product of ON and G.)

If u_{g} ∈ A_{g} ∩ hU(A), then A = {au_{g} | g ∈ G, a ∈ A_{1}}, and the multiplication is defined by

(au_{g})(bu)h) = a(u_{g}bu_{g}^{−1})α(g, h)u_{gh},

where α(g, h) = u_{g}u_{h}u_{gh}^{−1} ∈ U(A_{1}) and α: G × G → U(A_{1}) is a factor-set by the associativity condition.

In particular, if A_{1} = k, then α ∈ Z^{2}(G, k^{∗}) is a 2-cocycle, and another choice of the homogenous units provides a cocycle cohomologous to α. In this case, A is called a twisted group algebra, and we denote A = k^{α}G.

10.4. Modules over strongly graded algebras. Assume that A is strongly G-graded and let H, K be subgroups of G. Then A is an (A_{H}, A_{K})-bimodule, projective as a left A_{H} -module and as a right A_{K} -module, and we have that

A_{H} A_{A_{K}} = ⨁_{g∈[H\G/K]} A_{HgK} ≃ ⨁_{g∈[H\G/K]} A_{H} ⊗_{H∩^{g}K} A_{^{g}K}.

Further, if M, M′ are A-modules, then Hom_{A}(M, M′) is an OG-module as follows. Since A_{g}A_{g^{−1}} = A_{1} , there are subsets {a_{1}, …, a_{n}} ⊂ A_{g}, {a′_{1}, …, a′_{n}} ⊂ A_{g^{−1}} (called dual bases) such that

∑_{i=1}^{n} a_{i}a′_{i} = 1.

If ϕ ∈ Hom_{A_{1}}(M, M′), then define

gϕ(m) = ∑_{i=1}^{n} a_{i}ϕ(a′_{i}m) for all m ∈ M ,

(this being independent on the choice of the dual bases). In particular, End_{A}(M) is a G-algebra.

These observations show that the Nakayama relations, Higman’s Criterion and the Green correspondence hold for A-modules, simply by replacing OH with A_{H} for any subgroup H in G.

10.5. Graded homomorphisms. Assume that A is a G-graded algebra and M, M′ are G-graded A-modules. Then Hom_{A}(M, M′) has a natural G-grading as follows. For g ∈ G, M(g) is the G-graded A-module defined by M(g) = M, M(g)_{x} = M_{xg} for all x ∈ G. Then

Hom_{A}(M, M′) = ⨁_{g∈G} Hom_{A}(M, M′)_{g},

where

Hom_{A}(M, M′)_{g} = Hom_{A}(M, M′(g)) = Hom_{A}(M(g^{−1}), M′).

In particular, E = End_{A}(M)^{op} is a G-graded algebra and M becomes a G-graded (A, E)- bimodule (that is, A_{g}M_{x}E_{h} ⊆ M_{gxh} for all g, h, x ∈ G). Observe also that E is a crossed product if and only if M is G-invariant, that is, M ≃ M(g) in A-gr for all g ∈ G.

If A is strongly graded, then by Dade’s theorem 10.2

Hom_{A}(M, M′) ≃ Hom_{A_{1}}(M_{x}, M_{xg}) for all g, x ∈ G,

and M is G-invariant if and only if M_{1} ia G-invariant, that is, A_{g} ⊗_{A_{1}} M_{1} ≃ M_{1} in A_{1}-mod. We shall denote by

G_{M} = G_{M_{1}} = {g ∈ G | R_{g} ⊗_{A_{1}} M_{1} ≃ M_{1} in A_{1}-mod}

the stabilizer of M and M_{1} in G.

10.6. Modules lying over a simple A_{1} -module. If U is a simple A-module, then _{A_{1}}U clearly has a simple A_{1} -submodule S, and letting M = A⊗_{A_{1}} S, there is an epimorphism M → U in A-mod.

It is useful to consider the full subcategory (A | M)-mod of A-it modules lying over S, consisting of A-modules U for which there is an exact sequence

M^{(J)} → M^{(I)} → U → 0

for some finite sets I and J. It is not difficult to see that (A | M)-mod is an abelian subcategory of A-mod and M is a projective object of this category. Denoting E = End_{A}(M)^{op}, we have that the functors

Hom_{A}(M, −): (A | M)-mod → E-mod

and

M ⊗_{E} −: E-mod → (A | M)-mod

are inverse equivalences of categories.

Let H = G_{S} be the stabilizer of S. Then, by Schur’s lemma, E_{g} = 0 for g ∈ G \ H, so E = E_{H} ≃ k^{α}H is a twisted group algebra, since E_{1} ≃ End_{A_{1}}(S)^{op} ≃ k. Since the above equivalences also holds with A_{H} and M_{H} instead of A and M, one can easily deduce that

A ⊗_{A_{H}} −: (A_{H} | M_{H})-mod → (A | M)-mod

is an equivalence.

We can now describe the structure of A-modules lying over S. If U ∈ (A | M)-mod, then U has a unique simple A_{H} -submodule V such U ≃ A ⊗_{A_{H}} V . Moreover, _{A_{1}}U is semisimple, and

A_{1}U ≃ e( ⨁_{g∈[G/H]} A_{g} ⊗_{A_{1}} S),

where e is the k -dimension of the E -module corresponding to U (and to V ).

Let us discuss the stable case, that is, when G_{S} = G. Then we have the isomorphisms of functors

Hom_{A}(M, −) ≃ Hom_{A_{1}}(S, −) and N ⊗_{E} − ≃ S ⊗_{k} −.

It follows in particular that one can define a structure of an A-module on S extending the given structure if and only if k is a trivial k^{α}H -module, or quivalently, α is a coboundary. We say in this case that S extends to A.

10.7. Graded Jacobson radical. The graded Jacobson radical J_{gr}(A) of the G-graded algebra A is, by definition, the intersection on the maximal graded left ideals of A.

If A is strongly graded, Theorem 10.2 implies that J_{gr}(A) = J(A_{1})A. Further, since a simple A-module is a semisimple A_{1} -module, one deduces that J_{gr}(A) ⊆ J(A), and by Higman’s criterion, if |G| is invertible in O, then J_{gr}(A) = J(A).

Actually the last two statements are true even if A is not strongly graded, but some additional techniques are needed to prove this generalization.

10.8. Modules lying over an indecomposable A_{1} -module. If S is an indecomposable non-simple A_{1} -module, we need to redefine the category of A-modules lying over S. In this case let (A | M)-mod = add[M] be the full additive subcategory of A-mod consisting of summands of direct sums of copies of M.

Then the functors Hom_{A}(M, −) and M ⊗_{E} − are inverse equivalences between (A | M)-mod and the category of projective E -modules.

Denoting D = E/J_{gr}(E), then a variant of Fitting’s lemma shows that D = D_{H} = k^{α}H is a twisted group algebra for some α ∈ Z^{2}(G, k^{∗}), where H = G_{S}.

Reduction modulo J_{gr}(E) ⊆ J(E) is an additive functor inducing a bijection between the projective indecomposable E -modules and the projective indecomposable D-modules.

Since E_{H}/J_{gr}(E_{H}) ≃ D, we obtain the structure of an A-module U lying over S, as follows. U has an indecomposable A_{H} -module V (which is unique only up to isomorphism) such that U ≃ A ⊗_{A_{H}} V, and regarded as an A_{1} -module,

A_{1} ≃ e( ⨁_{g∈[G/H]} A_{g} ⊗_{A_{1}} S),

where e is the k -dimension of the D-module corresponding to U and to V .

10.9. Green’s indecomposability theorem. Assume that H = G_{S} is a p-group. Then H^{2}(G, k^{∗}) is trivial, so for any α ∈ Z^{2}(G, k^{∗}), k^{α}G ≃ kG is a local ring. Since k is the unique simple gG-module and kG is the unique projective indecomposable kG-module, by 11.6 and 11.8 we deduce:

a) If S is simple and U is a simple A-module lying over S, then S extends to A_{H} and U ≃ A ⊗_{A_{H}} S.

b) If S is indecomposable, then A ⊗_{A_{1}} S is an indecomposable A-module.

10.10. Blocks of normal subgroups. Let again A = OH regarded as a G-graded O-algebra, where N ⊴ H and G = H/N.

If e is a block idempotent of A_{1} , let b = A_{1}e be the corresponding block algebra,

T = {h ∈ H | heh^{−1} = e}

be the stabilizer of e in H and G_{b} = T/N. We see that G acts on Z(A_{1}) and that

f = ∑_{g∈[G/G_{b}]} gfg^{−1}

is a central idempotent of A. We say that the block idempotent ẽ of A covers e if ẽf ≠ 0.

10.11. Theorem. Let b = A_{1}e be a block of A_{1} and B = Aẽ a block of A.

a) The blocks of A_{1} covered by B are the (A_{1}, A_{1})-summands of B and are Gconjugated.

b) The blocks of A covering b are the (A, A)-bimodule summands of AeA = Af

c) eAe = eA_{G_{b}} is a strongly G_{b} -graded O-algebra, and it is Morita equivalent with Af.

d) If B covers b, then there is a defect group D̃ of B such that D̃ ∩ N is a defect group of b.

11. Characters and numerical invariants of blocks

Let (K, O, k) be a splitting p-modular system for G and we nay assume that K ⊂ ℂ. Then KG is semisimple and we shall discuss KG-modules in greater detail.

11.1. Definition and first properties of characters. Let V be a KG-module. Choosing a K-basis of V , we see that V affords a matrix representation X_{V} : G → M_{n}(K). For any g ∈ G, trace(X_{V} (g)) does not depend on the choice of the basis, so we may define the character χ_{V} of V by

χ_{V} : G → K, χ_{V} (g) = trace(X_{V} (g)).

Observe that the degree deg(χ_{V}) = χ_{V} (1) equals dim_{K}(V). Characters of degree 1 are called linear characters.

The characters afforded by simple KG-modules are called ireducible characters. Denote by Irr(G, K) = {χ_{1}, …, χ_{k}} the set of irreducible characters of G, where k = k(KG) (the notation of Section 3).

By the same property of the trace we have that χ_{V} (hgh^{−1}) = χ_{V} (g) for all g, h ∈ G. This means that χ_{V} is constant on the conjugacy classes of G, and we say that a character is a class function. We denote by Cl(G) the set of conjugacy classes of G, by F(G, K) the K-algebra of K-valued class functions on G, and notice that by Proposition 3.5 we have

k = | Irr(G, K)| = dim_{K} Z(KG) = | Cl(G)| = dim_{K} F(G, K).

Exercises. a) For every g ∈ G, χ_{V} (g) is a sum of |G|-th roots of unity. (Hint: use that X_{V} (g) is similar to a diagonal matrix.) Deduce that the values of characters are algebraic integers.

b) χ_{V} (g^{−1}) = \overline{χ_{V} (g)} (the complex conjugate of χ_{V} (g)).

The character table of G is the square matrix (χ_{i}(g_{j}))_{1≤i,j≤k} having algebraic integers as entries, where {χ_{1}, …, χ_{k}} = Irr(G, K) and {g_{1}, …, g_{k}} is a complete system of representatives for Cl(G). This matrix can be algorithmically computed, and its knowledge is important: it determines for instance the degrees of irreducible characters of G, the order of G and of the centralizers C_{G}(g) for g ∈ G, and the normal subgroups of G as unions of conjugacy classes.

11.2. Remarks. a) The character of the trivial KG-module K is χ_{K} = 1_{G} : g ↦ 1 for all g ∈ G.

b) The character ρ = χ_{KG} of the regular KG-module _{KG}KG is defined by ρ(1) = |G| and ρ(g) = 0 for g ≠ 1.

c) If V, W are KG-modules, then χ_{V ⊕W} = χ_{V} +χ_{W} , χ_{V ⊗W} = χ_{V} ·χ_{W} and χ_{V^{∗}} = χ̄_{V} .

d) If H is a subgroup of G and U is a KH -module with character χ, then the character χ^{G} of Ind_{H}^{G} U is given by

χ^{G}(g) = \frac{1}{|H|} ∑_{x∈G,x^{−1}gx∈H} χ(x^{−1}gx).

11.3. The scalar product. If ϕ, ψ ∈ F(G, K) are two class functions, define

(ϕ, ψ)_{G} = \frac{1}{|G|} ∑_{g∈G} ϕ(g)\overline{ψ(g)}.

Then (−, −) is a scalar product on F(G, K) and its importance is given by the following theorem.

11.4. Theorem. a) The set Irr(G, K) is an orthonormal basis of F(G, K). b) The KG-modules V and W are isomorphic if and only if χ_{V} = χ_{W} .

Exercises. a) If χ ∈ Irr(G, K), then ∑_{g∈G} χ(g) = \{ \begin{matrix}|G| & if χ = 1_{G} \\ 0 & otherwise \end{matrix} .

b) If ϕ = ∑_{i=1}^{k} a_{i}χ_{i} ∈ F(G, K), then (ϕ, ϕ)_{G} = ∑_{i=1}^{k} |a_{i}|^{2} . Consequently, if (ϕ, ϕ) = 1, then ϕ ∈ Irr(G, K).

c) ρ = ∑_{i=1}^{k} χ_{i}(1)χ_{i} and |G| = ∑_{i=1}^{k} χ_{i}(1)^{2}.

d) G is abelian if and only is every irreducible character of G is linear.

11.5. Characters in p-blocks. If B is a block of OG, then K ⊗_{O} B is a sum of blocks of KG, each containing exactly one simple KG-module. Their characters are said to belong to the p-block B. In his way we distributed the irreducible characters of G into blocks, and we denote by Irr(B) the number of irreducible characters belonging to B and let k(B) = | Irr(B)|. Let also ℓ(B) be the number of simple B -modules and f_{0}(B) the number of projective simple B -modules.

Assume that B has defect d and a defect group D of order p^{d}. We record the following numerical characterizations of the defect (n_{p} denoting the p-part of the integer n).

11.6. Theorem(Brauer). a) [G : D]_{p} is the largest power of p dividing χ(1)_{p} for all χ ∈ Irr(B).

b) Denoting |G|_{p} = p^{a}, then p^{2a−d} is the exact power of p dividing rank_{O} B.

It is also known that χ(1) divides [G : Z(G)] for all χ ∈ Irr(G, K). Let χ ∈ Irr(B). Dade introduced the defect def(χ) of χ by the equality

p^{def(χ)} = \frac{|G|_{p}}{χ(1)_{p}} ,

and denote by k_{δ}(B) the number of irreducible characters of defect δ in B. The height ht(χ) of χ is defined by

χ(1)_{p} = \frac{|G|_{p}}{d} p^{ht(χ)}.

Then B contains at least one character of height 0, and observe that k_{d}(B) is the number of characters of height 0 in B.

The subgroups of the form N_{G}(P), where P is a p-subgroup of G, are called plocal subgroups. According to Alperin [Al1], the main problem of modular representation theory is to describe the representations of G in terms of information coming from p-local subgroups. We end by presenting a list of conjectures which are part of this philosophy and originate from a famous list of problems of Richard Brauer. We refer to [Al3], [D3], [K], [M1], [M2] and [R] for further information and bibliography.

11.7. Alperin’s weight conjecture. If B is a block of OG, then

ℓ(B) = ∑_{Q,b} f_{0}(b),

where Q runs over the p-subgroups of G up to conjugacy and b over the Brauer correpondents of B in N_{G}(Q).

This formula implies that if B has positive defect, then ℓ(B) is p-locally determined.

11.8. Alperin-McKay conjecture. Let B be a block of OG with defect group D of order p^{d} and b the block of ON_{G}(D) corresponding to B via the Brauer correspondence. Then

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

(These two conjectures are proved in the T.I. situation by G. Michler and H. Blau and for p-solvable groups.)

11.9. Dade’s conjecture. Let S_{p}(G) be the set of of stricly increasing chains

C = (Q_{0} < Q_{1} < ⋯ < Q_{n})

of nontrivial p-subgroups of G and let |C| = n be the length of C. Then G acts by conjugation on S_{p}(G) and let

G_{C} = N_{G}(Q_{0}) ∩ ⋯ ∩ N_{G}(Q_{n})

be the stabilizer of C.

Then Dade’s conjecture states that if the maximal normal p-subgroup O_{p}(G) = 1 and B is a block of positive defect, then for all δ > 0,

∑_{C,b}(−1)^{|C|}k_{δ}(b) = 0,

where C runs over S_{p}(G) up to conjugacy and b over the Brauer correspondents of B in G_{C}.

This implies Alperin’s conjecture, and for δ = d, it gives the Alperin-Mckay conjecture. Note that the use of p-chains is due to R. Knörr and G. Robinson. Dade also formulated in [D3] several stronger forms of the conjecture, the strongest being compatible with Clifford theory and suitable to be reduced to the case of simple groups. These conjectures have been checked in many particular cases.

11.10. Brauer’s k(B)-problem. If B is a block with defect group D, is it true that k(B) ≤ D?

(This is open even for p-solvable groups.)

11.11. Brauer’s problem 21. Given the integer K, if B is a block with defect d, is it true that d is bounded in terms of k?

(It was proved in the case of p-solvable groups by B. Külshammer.)

11.12. Brauer’s height zero conjecture. The defect groups of B are abelian if and only if all irreducible characters in B have height zero.

(This conjecture is true for p-solvable groups (D. Gluck and T. Wolf) and in the T.I. situation.)

11.13. Robinson’s conjecture. If B has non-abelian defect group D, then

p^{ht χ} < [D : Z(D)]

for all χ ∈ Irr(B).

11.14. Olsson’s conjecture. If B has defect group D of order p^{d}, then

k_{d}(B) ≤ [D : D′].

(Open for p-solvable groups.)

References

Books.

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

[Be] D.J. Benson, Representations and cohomology I, Cambridge University Press, Cambridge, 1991. [C-R] C.W. Curtis and I. Reiner, Methods of representation theory I-II, Wiley, N.Y., 1981/1987.

[Fe] W. Feit, The Representation Theory of Finite Groups, North-Holland Amsterdam, 1982.

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

Survey papers.

[Al1] J.L. Alperin, The main problem of block theory, Proc. Conf. Finite Groups (Univ. Utah 1975), Academic Press, N.Y. (1976), 341-356.

[Al2] J.L. Alperin, Local representation theory, Proc. Symp. Pure Math. 37 (1980), 369-375.

[Al3] J.L. Alperin, Weights for finite groups, Proc. Symp. Pure Math. 47 (1986), 369-379.

[B] R. Brauer, Representations of finite groups, Lectures in Modern Mathematics, vol. 1, Wiley, New-York, 1963, pp. 133-175.

[D1] E.C. Dade, A Clifford theory of blocks, Proc. Symp. Pure Math. 21 (1971), 33-36.

[D2] E.C. Dade, The equivalence of various generalizations of group rings and modules, Math. Z. 181 (1982), 335-344.

[D3] E.C. Dade, Counting characters in blocks, 2.9, Proc. Conf. on Representation Theory (Univ. Ohio 1995).

[K] B. Külshammer, Modular representations of finite groups: Conjectures and examples, Akad. gemein. Wiss. Erfurt, Sitzungsber. Math.-Nat. Kl. 7 (1996), 93-125.

[M1] G.O. Michler, Modular representation theory and the classification of finite simple groups, Proc. Symp. Pure Math. 47 (1986), 223-232.

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[Z] A. Zimmermann, Structure of blocks with cyclic defect groups and Green correspondence, An. St. Univ. Constanţa IV(1) (1996), 24-73.

“Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Mihail Kogălniceanu 1, RO-3400 Cluj-Napoca, Romania

E-mail address: marcus@math.ubbcluj.ro