Bilinear forms on G-algebras
type: Item
instance of: scholarly article
title: Bilinear forms on G-algebras
zbMATH Open document ID: 0613.20009
DOI: 10.1016/0021-8693(86)90223-1
author: Michel Broué
author: Geoffrey R. Robinson
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Given a G-algebra A (in the sense of \textit{J. A. Green} [Math. Z. 107, 133-150 (1968; Zbl 0164.340)]), with a G-invariant linear form the authors construct bilinear forms on the ideal \(Tr^ G_ P(A^ P)\) of \(A^ G\), for each p-subgroup P of G. The rank of this form is computed, and special cases then yield information about the number of blocks with given defect group and the multiplicities of Scott modules as direct summands of modules of the form \(M\otimes M^*\) (cf., \textit{D. J. Benson} and \textit{J. F. Carlson} [J. Algebra 104, 329-350 (1986; Zbl 0612.20005)]). An extension of the multiplicity formula to the context of pointed groups is presented; it was communicated to the authors by L. Puig. A second part of the paper, nearly independent of the first part, discusses the connections between Hecke algebras and group algebras, with applications to fusion and the structure of permutation modules. At the end, the results are linked with those in the first part to obtain a new proof of the main result of \textit{G. R. Robinson}'s earlier paper [J. Algebra 84, 493-502 (1983; Zbl 0519.20011)] on the number of blocks with given defect group.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 3991756
zbMATH Keywords: G-algebra
zbMATH Keywords: G-invariant linear form
zbMATH Keywords: bilinear forms
zbMATH Keywords: rank
zbMATH Keywords: number of blocks
zbMATH Keywords: defect group
zbMATH Keywords: Scott modules
zbMATH Keywords: direct summands
zbMATH Keywords: pointed groups
zbMATH Keywords: Hecke algebras
zbMATH Keywords: group algebras
zbMATH Keywords: fusion
zbMATH Keywords: permutation modules
reviewed by: William H. Gustafson
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90223-1
OpenAlex ID: W2038201008
cites work: Nilpotent elements in the Green ring
cites work: On a Theorem of G. Robinson
cites work: On Scott Modules and p-Permutation Modules: An Approach through the Brauer Morphism
cites work: Brauer morphism between modular Hecke algebras
cites work: Some remarks on defect groups
cites work: Multiplicities, Scott Modules and Lower Defect Groups
cites work: Q3035669
cites work: Quelques remarques sur un thème de Knörr. (Some remarks on a theme of Knörr)
cites work: Pointed groups and construction of characters
cites work: The number of blocks with a given defect group
description: scientific article; zbMATH DE number 3991756