Block theory, branching rules, and centralizer algebras

Ellers, Harald; Murray, John

doi:10.1016/j.jalgebra.2003.12.029

Cited by 5 publications

(9 citation statements)

References 19 publications

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“…Nevertheless, there are also quite a few examples for which the answer to Question 5 is positive. The paper [6] shows that the answer is positive when G = S n and H = S n−1 . Computer calculations done by the first author using Magma [3] have shown that the answer is positive when G = S n and H = S m for all cases with m ≤ n ≤ 8.…”

Section: Question 5 Is the Center Of F G H Generated As An Algebra Bmentioning

confidence: 99%

“…It has a cyclic defect group, and 5 irreducible characters, labelled by the partitions [6], [4,2] Thus the Brauer tree is a straight line with 5 nodes and 4 edges. The only bit of information we need here is that Hom F G (P ( [6]), P ([2 2 , 1 2 ])) = 0, where P (λ) is the projective cover of the simple module D(λ). In other words, these PIM's have no composition factor in common.…”

Section: Examplementioning

confidence: 99%

“…Back to the displayed equalities. It's easy to see that F N ↑ G e = P ( [6]) and sgn N ↑ G e = sgn G ⊗ F N ↑ G e = P ([2 2 , 1 2 ]). Then by the last statement of the last paragraph, End ([2 2 , 1 2 ])) decomposes into a direct product of two non-zero F -algebras (each 2-dimensional, each commutative, with a 1-dimensional Jacobson radical).…”

Section: Examplementioning

confidence: 99%

“…The authors have been studying the representation theory of these algebras in several recent and not so recent papers [4], [5], [6], [7], [10], [11], mainly in cases where G is p-solvable and H is normal, or when G = S n and H = S m for n − 3 ≤ m ≤ n. Part of the original motivation was to see whether there might be a "weight conjecture" for these algebras, one that would simultaneously generalize Alperin's weight conjecture and Brauer's First Main Theorem on Blocks. This idea is explained in more detail in in [4], [5], and [6]. Also, when H is a p-subgroup these algebras play an important role in Green's approach to modular representation theory and in Puig's theory of points.…”

mentioning

confidence: 99%

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The centralizer of a subgroup in a group algebra

Danz

Ellers

Murray

2012

Proceedings of the Edinburgh Mathematical Society

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Let F be an algebraically closed field, G be a finite group and H be a subgroup of G. We answer several questions about the centralizer algebra FGH. Among these, we provide examples to show that•the centre Z(FGH) can be larger than the F-algebra generated by Z(FG) and Z(FH),•FGH can have primitive central idempotents that are not of the form ef, where e and f are primitive central idempotents of FG and FH respectively,•it is not always true that the simple FGH-modules are the same as the non-zero FGH-modules HomFH(S, T ↓ H), where S and T are simple FH and FG-modules, respectively.

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Section: Question 5 Is the Center Of F G H Generated As An Algebra Bmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

Section: Examplementioning

confidence: 99%

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confidence: 99%

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The centralizer of a subgroup in a group algebra

Danz

Ellers

Murray

2012

Proceedings of the Edinburgh Mathematical Society

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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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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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Centralizer Algebras and Degenerate Affine Hecke Algebras

Ellers

Murray

2013

Communications in Algebra

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Block theory, branching rules, and centralizer algebras

Cited by 5 publications

References 19 publications

The centralizer of a subgroup in a group algebra

The centralizer of a subgroup in a group algebra

Modular centralizer algebras corresponding to p-groups

Centralizer Algebras and Degenerate Affine Hecke Algebras

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