Endomorphism rings of permutation modules.
type: Item
instance of: scholarly article
title: Endomorphism rings of permutation modules.
zbMATH Open document ID: 1218.20005
author: Natalie Naehrig
published in: Journal of Algebra
publication date: 2010-11-18T00:00:00Z
review text: Let \(k\) be an algebraically closed field, let \(G\) be a finite group, and let \(P\) be a Sylow \(p\)-subgroup of \(G\). In this paper, the author studies the permutation module \(k_P^G=\mathrm{Ind}_P^G(k)\) and its endomorphism ring \(\mathfrak E=\mathrm{End}_{kG}(k_P^G)\). The paper is motivated by \textit{J. L. Alperin}'s suggestion to investigate his weight conjecture [Proc. Symp. Pure Math. 47, 369-379 (1987; Zbl 0657.20013)] through the endomorphism rings of certain permutation modules. The author focuses on the socle and head constituents of the indecomposable direct summands of \(k_P^G\) and of the projective indecomposable modules of \(\mathfrak E\). The computational methods utilize the computer algebra system GAP. The examples considered are: \(A_6\bmod 2\); \(S_6\bmod 3\); \(A_7\bmod 2,3\); \(S_7\bmod 2,3\); \(A_8\bmod 3\), \(A_9\bmod 2,3\); \(M_{11}\bmod 2,3,5,11\); \(M_{12}\bmod 2,3\); \(M_{22}\bmod 2\); \(U_3(3)\bmod 2\); \(J_2\bmod 2\); \(L_2(11)\bmod 2\); \(L_2(13)\bmod 2,3\); \(L_2(17)\bmod 2,3\); \(L_2(19)\bmod 2,3\).
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C40
zbMATH DE Number: 5817242
zbMATH Keywords: finite groups
zbMATH Keywords: permutation modules
zbMATH Keywords: endomorphism rings
zbMATH Keywords: Alperin weight conjecture
zbMATH Keywords: indecomposable direct summands
zbMATH Keywords: projective indecomposable modules
reviewed by: Frauke Bleher
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.12.011
OpenAlex ID: W1999172997
cites work: The alperin and dade conjectures for the simple mathieu groups
cites work: Weights for symmetric and general linear groups
cites work: Q3804896
cites work: Q4320290
cites work: The Alperin and Dade conjectures for the simple Held group
cites work: Brauer morphism between modular Hecke algebras
cites work: Q3922829
cites work: Blocks with cyclic defect groups
cites work: Counting characters in blocks. I
cites work: Q4240523
cites work: The blocks of finite general linear and unitary groups
cites work: Modular Harish-Chandra series, Hecke algebras and (generalized) \(q\)-Schur algebras
cites work: Q5688169
cites work: On a Theorem of H. Sawada
cites work: Weights and vertices for characters of \(\pi\)-separable groups
cites work: Some Remarks on a Conjecture of Alperin
cites work: Fusion category algebras.
cites work: Examining Green correspondents of weight modules
cites work: Modular Representations of Finite Groups with Unsaturated Split (B,N)-Pairs
DOI: 10.1016/J.JALGEBRA.2009.12.011
description: scientific article; zbMATH DE number 5817242