Defects of Irreducible Characters ofp-Soluble Groups

Abstract

We prove a refinement of the p-soluble case of Robinson's conjectural local characterization of the defect of an irreducible character.

Anchors of irreducible characters

“…The simple k ⊗ O OGe χ -module N ϕ corresponding to ϕ ∈ IBr(G|χ) has dimension ϕ(1). Thus we have an isomorphism of k-algebras (1) .…”

“…The map f : OG → OGe χ sending x ∈ OG to xe χ is an epimorphism of O-orders; in particular, the O-orders OGe χ and OG/Ker(f ) are isomorphic. Since we assume that K and k are splitting fields for all finite groups arising in this paper, we have e χ = χ (1) |G| g∈G χ(g −1 )g, and as a Kalgebra, KGe χ is isomorphic to the matrix algebra K χ(1)×χ (1) . In particular, KGe χ has up to isomorphism a unique simple left module M χ , and we have dim K (M χ ) = χ(1).…”

“…Green [7], it has a defect group. This is used in work of Barker [1] to prove a part of a conjecture of Robinson (cf. [24, 4.1, 5.1]) for blocks of finite p-solvable groups.…”

“…If G is p-solvable, then by the Fong-Swan theorem [4, §22], for any ϕ ∈ IBr(G) there is χ ∈ Irr(G) such that χ • = ϕ. The fact that anchors are centric is essentially proved in the proof of [1,Theorem] as an immediate consequence of results of Knörr [14], Picaronny-Puig [20], and Thévenaz [26]; see the proof of 3.7 below for details.…”

Vertices for characters of $p$-solvable groups

“…A consequence drawn by G. R. Robinson of the well-known conjectures on representation theory of groups by E. C. Dade and himself is that, given χ ∈ Irr(G), there always can be found a radical p-subgroup R of G which is "big" in a defect group of the p-block of χ, and which has an irreducible character η ∈ Irr(R) with d(χ) = d(η). For p-solvable groups this is now a fact (see Theorem 2 of [11], or [1] for a partial result). Our Theorem A, parts (e) and (f), gives a canonical choice for Robinson's predicted R and η: if (Q, δ) is a vertex of χ, it suffices to take R the radical closure of Q in G and η = δ R .…”