Modular centralizer algebras corresponding to p-groups

Abstract

We study the Loewy structure of the centralizer algebra kP Q for P a p-group with subgroup Q and k a field of characteristic p.Here kP Q is a special type of Hecke algebra. The main tool we employ is the decomposition kP Q = kC P (Q ) I of kP Q as a split extension of a nilpotent ideal I by the group algebra kC P (Q ).We compute the Loewy structure for several classes of groups, investigate the symmetry of the Loewy series, and give upper and lower bounds on the Loewy length of kP Q . Several of these results were discovered through the use of MAGMA, especially the general pattern for most of our computations. As a final application of the decomposition, we determine the representation type of kP Q .If G is a finite group with subgroup H and k a commutative ring with identity, then as in [7], the centralizer algebra kG H consists of all elements of kG that are invariant under the conjugation action of H . There have been several recent investigations into the representation theory of kG H in the papers [8][9][10][11]18,19]. In these papers, one of the motivating problems is the identification of the block idempotents of kG H for G a p-solvable group and H P G, or G = S n and H = S m . For P a p-group with subgroup Q and k a field of characteristic p, kP Q has no nontrivial idempotents, and therefore the questions one might ask concerning the structure and representation theory of kP Q have a somewhat different flavor than the study of the more general kG H . In particular, this paper explores the Loewy structure of kP Q and its representation type.Jennings proved in [14] a theorem that now bears his name and which allows us to the compute the radical layers of the group algebra kP for P a p-group using certain characteristic subgroups {κ i } of P . More precisely, we let κ 1 = P and inductively define κ n as the subgroup of P generated by [κ s , κ t ] whenever s, t < n and s + t n, along with all pth powers of elements from κ r whenever r < n and pr n. So κ 2 = Φ(P ) and each κ i /κ i+1 is an elementary abelian p-group. Let {x ij } s i j=1 be With the notation from Section 2, we know that J (kP Q ) = J ⊕ I where we write C = C P (Q ) and J = J (kC) for brevity. In computing J d for d > 1 it is useful to consider two separate questions: when is J I = I J ? and when is I 2 ⊆ J I ? Proposition 3.1. Let P be a p-group with subgroup Q and write k P Q = kC I. Then J I = I J if and only if Q satisfies the following condition: for all x ∈ P and all c ∈ C , there exists q ∈ Q such that [x, qc] ∈ C .Proof. To establish I J ⊆ J I it is enough to check that κ x (c −1 − 1) ∈ J I for κ x ∈ Ω and c ∈ C . We compute κ x c −1 − 1 = c −1 − 1 κ cxc −1 + (κ cxc −1 − κ x ) Since (c −1 − 1)κ cxc −1 ∈ J I we need κ cxc −1 − κ x ∈ J I . So let Ω = e i=1 Ω j be an orbit decomposition of Ω under the left action of C , so that I = kΩ j as kC -modules. Then J I =

Symmetry of Endomorphism Algebras

“…Since kP is local this yields ℓℓ(kP ) = |P |, and hence P is cyclic by application of Jennings' Theorem for p-groups; the final contradiction. This proof generalizes to centralizer algebras kP Q with P a p-group and Q ≤ P , by using the results from [1] on ℓℓ(kP Q ). We already saw in Lemma 4.1 one necessary condition for symmetry, when Λ is a not necessarily local algebra.…”

“…Then b 0 ≃ k Ḡ and b P 0 ≃ k Ḡ P where P acts on k Ḡ by conjugation. Since Ḡ is a p-group for which k Ḡ P is Frobenius, we know from [1] that P ≤ Z( Ḡ), and hence P is abelian.…”

“…Since k 2 = 1 we know that there is a maximal chain with (k 2 , 1) as its left-most element, and from the chain we obtain two directed paths in the diagram. In particular, since (k, k 1 ) appears in this chain with k 1 fixed, there is at most one k for which (2,1) and (n, 1) have maximal chains given by the following:…”

“…If G is a finite group with subgroup H and k a field, then the endomorphism algebra E G (k H ↑ G ) of the permutation kG-module k H ↑ G is referred to as a Hecke algebra. It is not known in general when E G (k H ↑ G ) is symmetric or quasi-Frobenius, but this condition was explored in [8], and proved useful in [11] where H was assumed to be a Sylow p-subgroup of G. Moreover, the centralizer algebras kG H for H a subgroup of G are Hecke algebras of the form E H×G (k ∆H ↑ H×G ) and have been recently studied in [6], [7], [4], and [1], and the author and J. Murray have been engaged in finding necessary and sufficient conditions guaranteeing symmetry of kG H . For H = G we have kG G = Z(kG) and in [10] it was established that Z(kG) is symmetric precisely when G is p-nilpotent with abelian Sylow p-subgroups.…”