DEFECTS OF IRREDUCIBLE CHARACTERS OF p-SOLUBLE GROUPS

Laurence Barker March 1997 Department of Mathematics, Bilkent University, 06533 Bilkent, Ankara, Turkey. e-mail: barker@fen.bilkent.edu.tr

ABSTRACT: We prove a refinement of the p-soluble case of Robinson’s conjectural local characterisation of the defect of an irreducible character.

1: Introduction

Some recent refinements of Alperin’s weight conjecture by Dade [3] and Robinson [10] indicate undiscovered riches in the rapport between Clifford theory and the local theory of group representations. The p-soluble case may be a good place to start looking at subproblems arising from these conjectures, because it is here that Clifford theory works best.

We fix a prime p, and a finite group G. Let ν_{p} denote the p-adic valuation on the rational numbers. Thus ν_{p}(z) = log_{p}z_{p} for any rational number z. When we speak of an irreducible character of G, we shall always be referring to an absolutely irreducible Frobenius character. Recall that the defect of an irreducible character χ of G is the natural number d(χ) := ν_{p}(|G|/χ(1)).

The principle of local theory — only a vague statement can even hope to encompass the plethora of sometimes conflicting approaches — is that the positive defect irreducible characters of G (and many other aspects of positive-defect p-blocks) can be related to “local” objects; objects which are in some way associated with nontrivial p-subgroups of G. In the context of Alperin’s conjecture, the normalisers of the non-trivial p-subgroups would have to be considered “local”. But in some sense (this point of view is taken in much of Puig’s work, say [8], [9]), the less our objects depend on G itself, and on the inclusions of the p-subgroups in G, the “more local” our objects are. It is in this sense that the following conjecture is “very local”. The conjecture, “locally” characteristing the defect of an irreducible character, is part of the consequence [10, 5.1] of the refinement [10, 4.1] of Alperin’s conjecture.

Conjecture 1.A: (Robinson) Let λ be an irreducible character of a central psubgroup L of G, and let χ be an irreducible character of G lying over λ and lying in a p-block with defect group D. Then there exists some P with C_{D}(P) ≤ P ≤ D, and an irreducible character η of P lying over λ such that d(χ) = d(η).

When L = 1, the Conjecture simplifies attractively, and still has interesting consequences as noted in [10]. However, both here and in [10], the full generality is needed for inductive arguments.

The p-soluble case probably follows already from some (as yet, incompletely published) work of Dade. Indeed, Dade [3, page 98] writes:

… we can use Clifford theory for a normal subgroup N of G to reduce theorems and conjectures to the case where G is simple. Eventually we shall perform such a reduction for a strengthened form of our conjecture.

Dade informs me that his reductions subsume a proof of the p-soluble case of the strengthened form of his conjecture. Meanwhile, Robinson [10, page 324] writes:

It would appear from the results of this paper and [3] that Conjecture 4.1 is equivalent to Dade’s weight conjecture 17.10 in [3], although at first sight, Conjecture 4.1 seems stronger.

In this note, a clearer and more direct proof of a refinement of the p-soluble case of the Conjecture shows that by no coincidence (at least in the p-soluble case) do the Conjecture and the theorem of Knörr, Picaronny, and Puig in [1, 1.1] both provide us with a subgroup P of a defect group D such that C_{D}(P) = Z(P).

Let κ be the field of fractions of a characteristic-zero, integrally closed, complete local commutative noetherian ring O whose residue field k has characteristic p. (For instance, O may be the completion of the integral closure of the p-adic completion of the rational integers.) We shall not repeat, here, G-algebra theory introduced in Puig [7], [9]; the books Külshammer [5] and Thévenaz [12] each contain all the terminology we shall need. Given a subgroup H ≤ G, and an interior G-algebra A, we write A^{H} for the H-fixed subalgebra of A, write Tr_{H}^{G} for the relative trace map A^{H} → A^{G}, and write A_{H}^{G} for the image of Tr_{H}^{G}. When A is O-free, we define κA := κ⊗_{O} A as an interior G-algebra over κ. The induction map from characters of H to characters of G, and the induction functor from interior H-algebras to interior G-algebras are both denoted by Ind_{H}^{G}.

Given an irreducible character χ of G, we write e_{χ} for the centrally primitive idempotent of κG fixing a simple κG-module X affording χ. Then the action of κGe_{χ} on X engenders an interior G-algebra isomorphism κGe_{χ} ≅ End_{κ}(X). Puig has suggested (personal communication) that the “local invariants” of χ be identified with the “local invariants” (such as the defect groups or source algebras) of the Ofree primitive interior G-algebra OGe_{χ} (a subring of κGe_{χ} ). We shall adapt this idea.

We must review a fairly well-known G-algebra-theoretic version of a Cliffordtheoretic construction; all the notation in the following discussion will be needed in the proof of the Theorem below. Further details, including a demonstration that the construction is both possible and essentially unique, may be found in (for instance) [2, Section 4]. Let R be a normal subgroup of G, let \underline{G} := G/R, let ψ be an irreducible character of R inertial in G, and let e := e_{ψ}, which is a central idempotent of both κR and κG. Given an irreducible κG-character χ, then χ lies over ψ if and only if e_{χ} = e_{χ} e.

We realise (κGe)^{R} as a twisted group algebra κ̃\underline{G} of \underline{G} over κ with \underline{G}-grading κ̃\underline{G} = ⨁_{\underline{g}∈\underline{G}} κ˜\underline{g}, where each ˜ \underline{g} ∈ (gκRe)^{R}. Let κ^{∗} denote the multiplicative group of κ. Let R̂ be the cyclic subgroup of κ^{∗} such that |R̂| is the order of the element of H^{2}(\underline{G}, κ^{∗}) associated with κ̃\underline{G}. By Schmid [11, 7.3], |R̂| divides the exponent of R. It is also well-known that |R̂| divides |G: R|. We insist the elements ˜ \underline{g} be chosen such that R̂ contains all the values of the factor set \underline{α} of \underline{G} given by ˜ \underline{g}˜\underline{h} = α(\underline{g}, \underline{h})˜\underline{gh} for \underline{g}, \underline{h} ∈ \underline{G}. Let

1 → R̂ → Ĝ → \underline{G} → 1

be the central extension determined up to equivalence by the condition that \underline{G} has a section \{ˆ\underline{g} : \underline{g} ∈ \underline{G}\} in Ĝ satisfying ˆ \underline{g}ˆ\underline{h} = \underline{α}(\underline{g}, \underline{h})ˆ\underline{gh}. The inclusion R̂ ↪ κ is an irreducible character ψ̂ of R̂. Let ê := e_{ψ}. Identifying κĜê with κ̃\underline{G} via the correspondence ˆ \underline{g}ê ↔ ˜ \underline{g}, the algebra isomorphism

σ_{G} : κ̃\underline{G} ⊗_{κ} κRe˜→κGe such that a ⊗ b ↦ ab,

may be rewritten as

σ_{G} : κĜê ⊗_{κ} κRe˜→κGe such that ˆ \underline{g}ê ⊗ b ↦ ˜ \underline{g}b.

Let α be the factor set of G such that α(g, h) := \underline{α}(\underline{g}, \underline{h})^{−1} for g, h ∈ G, where \underline{g}, \underline{h} denote the images of g, h, respectively, in \underline{G}. Let κ̃G be a twisted group algebra with G-grading κ̃G = ⨁_{g∈G} κg̃ and such that each g̃h̃ = α(g, h)g̃h. It is easy to show that ˜ \underline{g}^{−1}g˜\underline{h}^{−1}h = α(g, h)˜\underline{gh}^{−1}gh. The algebra map θ : κ̃G → κRe given by g ↦ ˜ \underline{g}^{−1}g satisfies ˜ \underline{g}θ(g̃) = ge = θ(g̃)˜\underline{g}. Regarding κĜê ⊗_{κ} κRe as an interior G-algebra with structural map g ↦ ˆ \underline{g}ê ⊗ θ(g̃), then σ_{G} is an interior G-algebra isomorphism.

The Fong correspondence χ ↔ χ̂ between the irreducible characters χ of G lying over ψ, and the irreducible characters χ̂ of Ĝ lying over ψ̂ is characterised by the condition σ_{G}(e_{χ} ⊗ e) = e_{χ}. We also note that if R̂ ≤ Ĥ ≤ Ĝ and R ≤ H ≤ G such that Ĥ/R̂ = H/R, then σ_{G} restricts to an interior H-algebra isomorphism σ_{H} : κĤê ⊗_{κ} κRe˜→κHe.

For a normal subgroup K of G, an irreducible character μ of K, and a natural number d, let k(G, d, μ) denote the number of irreducible characters of G with defect d lying over μ.

Lemma: Let K be a normal subgroup of p-power index in G. Given a Sylow psubgroup P of G, a Sylow p-subgroup L of K central in G, an irreducible character μ of K inertial in G, and an irreducible character λ of L lying under μ, then k(G, d, μ) = k(P, d, λ).

PROOF: Letting R := O_{p′}(G), then K = LR, and μ = λ ⊗ ψ for some irreducible character ψ of R inertial in G. In the notation above, R̂ is trivial, and we have evident isomorphisms κP ≅ κĜê and κL ≅ κLe = κK̂ê. Thus σ_{G} and σ_{K} may be regarded as isomorphisms κP ⊗ κRe ≅ κGe and κL ⊗ κRe ≅ κKe, respectively. Since σ_{G}(e_{λ} ⊗ e) = σ_{K}(e_{λ} ⊗ e) = e_{μ}, the Fong correspondence χ ↔ χ̂ restricts to a defect-preserving bijective correspondence between the irreducible characters χ of G lying over μ, and the irreducible characters χ̂ of P lying over λ. ⊓ ⊔

The proof of the Lemma can be recast using the methods in Isaacs [4, Chapter 13]: ψ extends uniquely, as in [4, 13.3], to an irreducible character ψ̃ of G such that the matrices representing the elements of P have determinant unity. It can be shown that the condition χ̃ = χ̂ ⊗ ψ̃ specifies a defect-preserving bijective correspondence between the irreducible characters χ̃ of G lying over μ, and the irreducible characters χ̂ of P lying over λ.

Theorem: Suppose that G is p-soluble. Let λ be an irreducible character of a central p-subgroup L of G, and let χ be an irreducible character of G lying over λ and lying in a p-block with defect group D. Then there exists some P with C_{D}(P) ≤ P ≤ D, an irreducible character η of P lying over λ such that d(χ) = d(η), and an O-free primitive interior G-algebra A with defect group P such that κA ≅ κGe_{χ}.

PROOF: First note that, given any O-free interior G-algebra A such that κA ≅ κGe_{χ}, then A^{G} ≅ O, and in particular, A must be primitive. Furthermore, Picaronny– Puig [6, Proposition 1], the theorem of Knörr, Picaronny, and Puig in [1, 1.1], and the transitivity property of defect groups together imply that any such A has a defect group P satisfying C_{D}(P) ≤ P ≤ D. In particular, since O_{p}(G) ≤ D, we have O_{p}(Z(G)) ≤ P. It suffices to prove that there exists an O-free interior G-algebra A such that κA ≅ κGe_{χ}, and an irreducible character η of a defect group P of A such that η lies over λ, and d(χ) = d(η). No generality in lost in assuming that L = O_{p}(Z(G)).

We argue by a double induction, firstly on |G: Z(G)|, and secondly on |G|. We may assume that G is non-abelian, because otherwise we put A = OGe_{χ}, whereupon we can let P be the Sylow p-subgroup of G, and put η = Res_{P}^{G}(χ). Since O_{p}(G)O_{p′}(G) ≰ Z(G), we can choose R such that R ≰ Z(G) and either R = O_{p}(G) or R = LO_{p′}(G). Let ψ be an irreducible character of R lying under χ, and let T be the inertia group of ψ in G. Clearly, L ≤ R and ψ lies over λ.

Suppose that T ≠ G. As is well-known, there exists a unique irreducible character χ′ of T lying over ψ such that χ = Ind_{T}^{G}(χ′). By induction, we may assume that there exists an O-free interior T -algebra A′ such that κA′ ≅ κT e_{χ′}, and an irreducible character η of a defect group P of A′ such that η lies over λ, and d(χ′) = d(η). Putting A := Ind_{T}^{G}(A′), then κA ≅ Ind_{T}^{G}(κA′) ≅ κGe_{χ}. A Mackey decomposition argument shows that A has defect group P. We have d(χ) = d(η) because χ(1) = |G:T |χ′(1). The assertion is now proved in the case T ≠ G.

Henceforth, we assume that T = G, and again consider the central extension Ĝ of \underline{G} by R̂. The cyclic group R̂ is the direct product R̂ = L̂L̂′ of a p-subgroup L̂ and a p′-subgroup L̂′. The linear character ψ̂ of R̂ decomposes as a tensor product ψ̂ = λ̂ ⊗ λ̂′ of linear characters λ̂, λ̂′ of L̂, L̂′, respectively. Let χ̂ be the irreducible character of Ĝ lying over ψ̂ such that σ_{G}(e_{χ} ⊗ e) = e_{χ}. Then χ̂ lies over λ̂. Now, if R = LO_{p′}(G), then Z(G) < R, hence |Ĝ : Z(Ĝ)| ≤ |G : R| < |G : Z(G)|. On the other hand, if R = O_{p}(G), then R̂ = O_{p}(Ĝ), hence Z(Ĝ) = R̂.O_{p′}(Z(Ĝ)), and |Ĝ:Z(Ĝ)| ≤ |\underline{G}:Z(\underline{G})| < |G:Z(G)|. Induction allows us to assume that there exists an O-free interior Ĝ-algebra Â such that κÂ ≅ κGe_{χ}, and an irreducible character η̂ of a defect group P̂ of Â such that η̂ lies over λ̂, and d(χ̂) = d(η̂).

The O-linear span Θ of {θ(g̃) : g ∈ G} in κRe is an algebra over O because the values of α are elements of O. For all g, h ∈ G, we have

hθ(g̃)h^{−1} = ˜ \underline{h}θ(h̃)θ(g̃)θ(h̃)^{−1} ˜\underline{h}^{−1} = α(h, h^{−1})^{−1}α(h, g)α(hg, h^{−1})θ(^{h̃}g)

which is an O-multiple of θ(^{h̃}g). So Θ aquires the structure of a G-algebra by restriction from κRe. For all x ∈ R, we have θ(x̃) = xe, so ORe ⊆ Θ ⊂ κRe. We define A := Â ⊗_{O} Θ, and write g1_{A} = ˜ \underline{g}1_{A} ⊗ θ(g̃) to specify the way in which we regard A as an interior G-algebra. Then κA ≅ κÂ ⊗_{κ} κRe ≅ κGe_{χ} as interior G-algebras.

We have A^{R} = Â ⊗ e. More generally, given R̂ ≤ Ĥ ≤ Ĝ and R ≤ H ≤ G such that Ĥ/R̂ = H/R, then A^{H} = Â_{H} ⊗ e. For all a ∈ Â_{H}, we have Tr_{G} ; _{H}(a) ⊗ e = Tr_{H}^{G}(a ⊗ e), so A_{H}^{G} = Â_{G} ; _{H} ⊗ e. In particular, 1_{A} ∈ A_{H}^{G} if and only if 1_{A} ∈ Â_{G} ; _{H}. We deduce that A has a defect group P satisfying P̂R̂/R̂ = P R/R.

Suppose that R = LL′, where L′ := O_{p′}(G). By comments in the first paragraph of the argument, L̂ ≤ O_{p}(Z(Ĝ)) ≤ P̂ and L ≤ P. So P̂L̂′/R̂ = P L′/R. Since η̂ lies over λ̂, the irreducible character γ̂ := η̂⊗λ̂′ of P̂L̂′ lies over ψ̂. Let γ be the irreducible character of P L′ lying over ψ such that σ_{P L′}(e_{γ} ⊗ e) = e_{γ}. Since γ(1)/γ̂(1) = ψ(1) = χ(1)/χ̂(1) and |G: P L′| = |Ĝ: P̂L̂′|, we have d(χ) − d(χ̂) = d(γ) − d(γ̂). Also, ν_{p}(|P̂L̂′|) = ν_{p}(|P̂|) and ν_{p}(γ̂(1)) = ν_{p}(η̂(1)), so d(γ̂) = d(η̂) = d(χ̂), hence d(χ) = d(γ). By the Lemma, there exists an irreducible character η lying over λ such that d(η) = d(γ). This completes the argument in the case R = LL′.

We may now assume that R = O_{p}(G). Then R̂ is a p-group, hence R̂ = L̂ ≤ P̂ and ψ̂ = λ̂. We claim that R ≤ P. Choosing an element c ∈ A^{P} such that Tr_{P}^{G}(c) = 1_{A}, let us write c = ∑_{i} a_{i} ⊗ b_{i} with each a_{i} ∈ Â, each b_{i} ∈ Θ, and the elements a_{i} linearly independent. For all g ∈ G, we have ^{g}(a_{i} ⊗b_{i}) = ^{g}a_{i} ⊗^{g}b_{i}. Each a_{i} is fixed by R ∩ P, so by the linear independence of the a_{i}, each b_{i} is fixed by R ∩ P. Mackey decomposition gives

∑_{RgP ⊆G} ∑_{i}^{g}a_{i} ⊗ Tr_{R∩^{g}P}^{R} (^{g}b_{i}) = ∑_{RgP ⊆G} Tr_{R∩^{g}P}^{R} (^{g}c) = 1_{A}.

Since Θ^{R} = Oe, at least one of the terms Tr_{R∩^{g}P}^{R} (^{g}b_{i}) must be an O^{∗}-multiple of e. Then e ∈ Θ_{R∩^{g}P}^{R}. For each h ∈ G, the elements h and θ(h̃) have the same conjugation action on Θ. In particular, θ(h̃)J(OR)e = J(OR)eθ(h̃), so J := J(OR)Θ is an ideal of Θ. The quotient Θ/J inherits, from Θ, the structure of an interior R-algebra over k, and 1_{Θ/J} ∈ (Θ/J)_{R∩^{g}P}^{R}. But each element of R maps to 1_{Θ} modulo J. So the conjugation action of R on Θ/J is trivial. Hence R ∩^{g}P = R, and the claim holds. Therefore P̂/R̂ = P/R.

Let η be the irreducible character of P lying over ψ such that σ_{P} (e_{η} ⊗ e) = e_{η}. Then η lies over λ. Since η(1) = η̂(1)ψ(1) and χ(1) = χ̂(1)ψ(1), we have

d(η) − d(η̂) = ν_{p}(|P |) − ν_{p}(|P̂|) + ν_{p}(ψ(1)),

d(χ) − d(χ̂) = ν_{p}(|G|) − ν_{p}(|Ĝ|) + ν_{p}(ψ(1)).

But |G:P | = |Ĝ:P̂| and d(χ̂) = d(η̂), hence d(χ) = d(η). ⊓ ⊔

ACKNOWLEDGEMENT

I wish to thank the referee for the short proof of the Lemma, and for pointing out some redundancies in an earlier version of this paper.

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