Cliques of irreducible representations ofp-solvable groups and a theorem analogous to Alperin’s conjecture

Ellers, Harald

doi:10.1007/bf02571963

Cited by 5 publications

(6 citation statements)

References 9 publications

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“…If this theorem had been available at the time [9] was written, it would have simplified considerably the induction proof of the main theorem in that paper.…”

Section: Theorem 11 Let K Be An Algebraically Closed Field Of Charamentioning

confidence: 99%

“…Let k be an algebraically closed field of characteristic p and let G be a finite group. The paper [9] introduced for each normal subgroup H of G a. If we imagine starting //off equal to 1 and "moving" //up along a chain of normal subgroups of G, we should see the blocks of G gradually emerge as we look at successively coarser partitions.…”

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confidence: 99%

“…In [9], it was conjectured that for each//, the number of//-cliques of irreducible k[G]modules is given by the sum Eg #(0, where Q runs through a set of representatives for the G-conjugacy classes of/7-subgroups of G, and where, for each Q, a(Q) is the number of (HnNciQ)) -cliques of irreducible k[Nc(Q)]-modules that contain only modules with vertex Q. When H = 1 this formula is Alperin's Conjecture [1]; when H = G this formula is essentially Brauer's First Main Theorem on Blocks.…”

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confidence: 99%

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Cliques of Irreducible Representations, Quotient Groups, and Brauer's Theorems on Blocks

Ellers¹

1995

Can. j. math.

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“…If this theorem had been available at the time [9] was written, it would have simplified considerably the induction proof of the main theorem in that paper.…”

Section: Theorem 11 Let K Be An Algebraically Closed Field Of Charamentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Cliques of Irreducible Representations, Quotient Groups, and Brauer's Theorems on Blocks

Ellers¹

1995

Can. j. math.

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“…This paper is one in a series [8][9][10][11] exploring the algebra kG H , the centralizer in the group algebra kG of the subalgebra kH , where k is a field of characteristic not zero, G is a finite group, and H is a subgroup of G. All these papers search for theorems similar to Alperin's weight conjecture [2]. The immediate goal is to find results that relate information about blocks of kG H or simple modules over kG H to p-local information.…”

Section: Introductionmentioning

confidence: 98%

Block theory, branching rules, and centralizer algebras

Ellers

Murray

2004

Journal of Algebra

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“…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 .…”

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confidence: 99%

Modular centralizer algebras corresponding to p-groups

Allan

2011

Journal of Algebra

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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 =

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Cliques of irreducible representations ofp-solvable groups and a theorem analogous to Alperin’s conjecture

Cited by 5 publications

References 9 publications

Cliques of Irreducible Representations, Quotient Groups, and Brauer's Theorems on Blocks

Cliques of Irreducible Representations, Quotient Groups, and Brauer's Theorems on Blocks

Block theory, branching rules, and centralizer algebras

Modular centralizer algebras corresponding to p-groups

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