Vertices for characters of $p$-solvable groups

Abstract

Abstract. Suppose that G is a finite p-solvable group. We associate to every irreducible complex character χ ∈ Irr(G) of G a canonical pair (Q, δ), where Q is a p-subgroup of G and δ ∈ Irr(Q), uniquely determined by χ up to Gconjugacy. This pair behaves as a Green vertex and partitions Irr(G) into "families" of characters. Using the pair (Q, δ), we give a canonical choice of a certain p-radical subgroup R of G and a character η ∈ Irr(R) associated to χ which was predicted by some conjecture of G. R. Robinson.

Anchors of irreducible characters

“…Let (W, γ) be a nucleus of χ such that Q is a Sylow p-subgroup of W and δ = Res W Q (α), where γ = αβ, with α a p ′ -special character and β a p-special character of W (cf. [17,Sections 2,3]). By Lemma 6.3, the trivial Brauer character is a constituent of β • .…”

“…Since |G/N | = p, it follows that χ is an irreducible character of G. Now, sinceβ is not G-stable, it is easy to see that χ is not p-factorable. On the other hand, N is a maximal normal subgroup of G. Thus (N,αβ) is a nucleus of G in the sense of [17], and the Sylow p-subgroups of N are the first components of the Navarro vertices of χ.…”

“…In [17] Navarro associated, via the theory of special characters, to each ordinary irreducible character χ of a p-solvable group G, a G-conjugacy class of pairs (Q, δ), where Q is a p-subgroup of G and δ is an ordinary irreducible character of Q, which behave in certain ways as the Green vertices of indecomposable modules (see also [5], [2] and [3]). We call such a pair (Q, χ) a Navarro vertex of χ.…”

Anchors of irreducible characters

“…Let (W, γ) be a nucleus of χ such that Q is a Sylow p-subgroup of W and δ = Res W Q (α), where γ = αβ, with α a p ′ -special character and β a p-special character of W (cf. [17,Sections 2,3]). By Lemma 6.3, the trivial Brauer character is a constituent of β • By Lemma 6.2, there exists an OW -lattice X affording β and with vertex Q.…”

Lifts and vertex pairs in solvable groups

“…We make use of the normal nucleus constructed by Navarro in [19]. We quickly summarize this construction.…”