THE DIMENSION OF A PRIMITIVE INTERIOR G-ALGEBRA

by LAURENCE BARKER

ABSTRACT: We give the residue class, modulo a certain power of p, for the dimension of a primitive interior G-algebra, in terms of the dimension of the source algebra. To illustrate, we improve a theorem of Brauer on the dimension of a block algebra.

Almost always, the G-algebras arising in group representation theory have been interior. Both in applications and in the general theory, it often suffices to consider primitive interior G-algebras. One of the themes of the theory is the characterisation of a primitive interior G-algebra in terms of its source algebra S. Stories revolving around this theme are told in the two books devoted to G-algebra theory — Külshammer [8], Thévenaz [15] — and in the papers — we mention particularly Puig [11], [12] — listed in their bibliographies. These stories focus on rich algebraic relationships between A and S; for a start, [11, 3.5] tells us that A and S are Morita equivalent. However, many outstanding conjectures, some old, some new, hark back to Brauer’s more arithmetical approach to group representation theory. See, for instance, conjectures in Alperin [1], Dade [4], Feit [6, Section 4.6], Robinson [13]. In this note, we point out an arithmetic relationship between A and S. As an illustrations, we shall discuss a theorem of Knörr on the dimension of a simply defective module, and shall improve a theorem of Brauer on the dimension of a block algebra.

Our notation is as in Thévenaz [15]; we repeat a little of it to set the scene, and extend it slightly. Let O be a complete local noetherian ring with an algebraically closed residue field k of prime characteristic p. Let G be a finite group, and let A be an interior G-algebra; as usual, we assume that A is finitely generated over O, and either free over O or annihilated by J(O). Given a pointed group H_{β} on A, we choose and element j ∈ β, and define A_{β} := jAj as an interior H-algebra. Now let X be an A-module; again we assume that X is finitely generated over O, and either free over O or annihilated by J(O). We define X_{β} := jX as an A_{β}-module. It is easy to extend the use of embeddings in Puig [12, 2.13.1] to show that X_{β} is unique up to a natural isomorphism of A_{β}-modules.

Henceforth, let us assume that A is primitive. Let P_{γ} be a defect pointed group on A. The source algebra A_{γ} associated with P_{γ} is an interior P -algebra. The multiplicity module V (γ) associated with P_{γ} is a projective indecomposable k_{∗}ˆ\overline{N}(P_{γ})-module. By the construction of V (γ), if 1_{A} = ∑_{t∈T} t as a sum of mutually orthogonal primitive idempotents of A^{P}, then dim_{k}V (γ) = |γ ∩ T |.

When V (γ) is simple, we say that A is simply defective. This notion has its origins in Knörr [7], and was introduced explicitly in Picaronny–Puig [10]. Necessary and sufficient conditions for A to be simply defective are to be found in [2, 1.3], [10, Proposition 1], and Thévenaz [14, 15, 9.3]. We recall that any block algebra of G over O or over k is simply defective. Also, the linear endomorphism algebras of certain OG-modules are simply defective (see below). Whenever A is simply defective, the p-part of the dimension of the multiplicity module is

(dim_{k}V (γ))_{p} = |N_{G}(P_{γ}):P |_{p}.

We shall give a formula for the residue class, modulo a certain power of p, for the O-rank rk_{O}A (interpreted as the k-dimension dim_{k}A when J(O) annihilates A). The terms of the formula are dim_{k}V (γ), some group-theoretic invariants of A, and a residue class of rk_{O}A_{γ}. Information about dim_{k}V (γ) and the grouptheoretic invariants is usually much easier to obtain than information about rk_{O}A_{γ}, so the formula may be seen as a congruence relation between rk_{O}A and rk_{O}A_{γ}. Since A_{γ} and V (γ) are uniquely determined up to a G-conjugacy condition, dim_{k}V (γ) and rk_{O}A_{γ} are isomorphism invariants of A. Similarly, given an A-module X, then rk_{O}X_{γ} is an isomorphism invariant of X.

For a p-subgroup P ≤ G, we define the spire of P in G to be

spr_{G}(P) := \{ \begin{matrix}min\{|P:P ∩ ^{g}P |\} & if P ≰G, \\ 0 & if P ≤ G. \end{matrix}

We interpret congruences modulo zero as equalities; this convention will apply to our results when P ≤ G.

PROPOSITION 1. Let A be a primitive interior G-algebra, let P_{γ} be a defect pointed group on A, and let X be an A-module. Then

rk_{O}X ≡ |G:N_{G}(P_{γ})|.dim_{k}V (γ).rk_{O}X_{γ} modulo |G:P |_{p} spr_{G}(P).

In particular, if A is simply defective, then

(rk_{O}X)_{p} ≡ (|G:P |.rk_{O}X_{γ})_{p}. modulo |G:P |_{p} spr_{G}(P).

Proof: If P ≤ G, then the points of P on A are precisely the G-conjugates of γ. Writing 1_{A} = ∑_{t∈T} t as above, we have

rk_{O}X = ∑_{gN_{G}(P_{γ})⊆G} |T ∩ ^{g}γ|.rk_{O}X_{(^{g}γ)} = |G:N_{G}(P_{γ})|.dim_{k}V (γ).rk_{O}X_{γ}.

Now suppose that P ≰G. Let H := N_{G}(P). By the Green Correspondence Theorem in Thévenaz [15, 20.1], there exists a unique point β of H on A such that P_{γ} ≤ H_{β}. Furthermore, β has multiplicity unity, that is to say, if 1_{A} = ∑_{s∈S} s as a sum of mutually orthogonal primitive idempotents of A^{H}, then precisely one element of S belongs to β.

Consider the induced interior G-algebra A′ := Ind_{H}^{G}(A_{β}). Recall that A′ = OG⊗_{OH} A_{β} ⊗_{OH} OG as OG−OG-bimodules, and A′ ≅ Mat_{|G:H|}(A_{β}) as algebras. Let X′ := OG⊗_{OH}X_{β} as an A′-module. Let γ′ and β′ be the points of P and H on A′ corresponding to γ and β, respectively. Since P_{γ′} is a defect pointed subgroup of H_{β′}, the Green Correspondence Theorem implies that there exists a unique point α′ of G on A satisfying P_{γ′} ≤ G_{α′}. Furthermore, α′ has multiplicity unity. By Puig [11, 3.6], A′_{α′} ≅ A as interior G-algebras, and via this isomorphism, X′_{α′} ≅ X as A-modules. A routine application of Mackey Decomposition and Rosenberg’s Lemma shows that if Q_{δ′} is a local pointed group on A′ not Gconjugate to P_{γ′}, then Q is contained in the intersection of two distinct Gconjugates of P. Therefore, every point of G on A′ distinct from α′ has a defect group contained in P ∩ ^{g}P for some g ∈ G − H. By Green’s Indecomposibility Criterion, |G: P |_{p} spr_{G}(P) divides rk_{O}X′ − rk_{O}X. We also have rk_{O}X′ = |G: H|rk_{O}X_{β}, and by the first paragraph of the argument,

rk_{O}X_{β} = |H:N_{G}(P_{γ})|.dim_{k}V (γ).rk_{O}X_{γ}.

⊓ ⊔

To illustrate Proposition 1, let us consider an indecomposable OG-module M (finitely generated over O, and either free over O or annihilated by J(O) ). Let P be a vertex of M, let U be a source OP -module of M, let F be the inertia group of U in N_{G}(P), and let m be the multiplicity of U as a direct factor of the restricted OP -module of M. The linear endomorphism algebra End_{O}(M) (interpreted as End_{k}(M) when J(O) annihilates M) is a primitive interior G-algebra with a defect pointed group P_{γ} such that M_{γ} ≅ U. Also, N_{G}(P_{γ}) = F, and dim_{k}(V (γ)) = m. By [2, 1.4], End_{O}(M) is simply defective if and only if m is the multiplicity of M in the induced OG-module of U. When these equivalent conditions hold, we say that M is simply defective. If M satisfies the hypothesis of Knörr [7, 4.5] (in particular, if M is an irreducible OG-module or a simple kG-module), then by Picarronny–Puig [10, Proposition 1], M is simply defective. Propsition 1 implies:

COROLLARY 2: Let M be an indecompoosable OG-module. Then, with the notation above,

rk_{O}M ≡ |G:F |.m.rk_{O}U modulo |G:P |_{p} spr_{G}(P).

In particular, if M is simply defective, then

(rk_{O}M)_{p} ≡ (|G:P |.rk_{O}U)_{p}. modulo |G:P |_{p} spr_{G}(P).

The rider to Corollary 2 relates to [7, 4.5] and [10, Proposition 3], but has slightly weaker hypothesis and conclusion.

LEMMA 3. Let G and H be finite groups. Let P_{γ} and Q_{δ} be defect pointed groups on, respectively, a primitive G-algebra A and a primitive H-algebra B. Then γ ⊗ δ is contained in a local point ϵ of P × Q on A ⊗_{O} B, and (P × Q)_{ϵ} is a defect pointed group on the primitive G × H-algebra A ⊗ B.

Proof: It is easy to check that A⊗B is primitive, and that γ⊗δ is contained in a point ϵ of P × Q. By considering the evident isomorphism of Brauer quotients

\overline{A}(P) ⊗ \overline{B}(Q) ≅ \overline{A ⊗ B}(P × Q)

we see that ϵ is local. On the other hand,

1_{A⊗B} ∈ Tr_{P ×Q}^{G×H}(A^{P} ⊗ B^{Q}.ϵ.A^{P} ⊗ B^{Q})

so (P × Q)_{ϵ} is a defect pointed group. ⊓ ⊔

THEOREM 4. Given a defect pointed group P_{γ} on a primitive interior Galgebra A, then

rk_{O}A ≡ (|G:N_{G}(P_{γ})|.dim_{k}V (γ))^{2}rk_{O}A_{γ} modulo |G:P |_{p}^{2} spr_{G}(P).

In particular, if A is simply defective, then

(rk_{O}A)_{p} = (|G:P |^{2}.rk_{O}A_{γ})_{p} modulo |G:P |_{p} spr_{G}(P).

Proof: This follows from Proposition 1 and Lemma 3 upon considering A as an A ⊗_{O} A^{op}-module by left-right translation. ⊓ ⊔

Let us consider a block idempotent b of OG with defect group P. Brauer [3, Theorem 1] used character theory to prove that the block algebra OGb satisfies

(rk_{O}OGb)_{p} = (|G||G:P |)_{p}.

A module-theoretic demonstration was later given by Michler [9, 2.1], and the result is generalised in Picaronny–Puig [10, Proposition 3]. Since OGb is simply defective, Theorem 4 gives, more precisely:

COROLLARY 5. Let b be a block idempotent of OG. Let (P, e) be a maximal Brauer pair associated with b, let T denote the inertia group of e in N_{G}(P), and let W be a copy of the isomorphically unique simple kC_{G}(P)e-module. Then

rk_{O}OGb ≡ (|G|dim_{k}W)^{2}|Z(P)|/|T ||C_{G}(P)| modulo (|G||G:P |)_{p} spr_{G}(P).

Proof. By an easy adaptation of part of the argument in Michler [9, 2.1], we may and shall assume that P ≤ G. Thévenaz [15, 40.13] describes a defect pointed group P_{γ} on OGb associated with (P, e), and also informs us that T = N_{G}(P_{γ}) and dim_{k}W = dim_{k}V (γ). By Puig [12, 6.6, 14.6],

rk_{O}(OGb)_{γ} = |N_{G}(P_{γ}):P C_{G}(P)||P | = |T ||Z(P)|/|C_{G}(P)|.

⊓ ⊔

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DEPARTMENT OF MATHEMATICS, BILKENT UNIVERSITY, 06533 BILKENT, ANKARA, TURKEY.

e-mail: barker@fen.bilkent.edu.tr

May 1997.