Eigenvector matrices of Cartan matrices for finite groups.
type: Item
instance of: scholarly article
title: Eigenvector matrices of Cartan matrices for finite groups.
zbMATH Open document ID: 1119.20014
author: Tomoyuki Wada
published in: Journal of Algebra
publication date: 2007-04-18T00:00:00Z
review text: Let \(p\) be a prime, let \(G\) be a finite group, and let \((K,R,F)\) be a \(p\)-modular system which is sufficiently large for \(G\). Moreover, let \(B\) be a block of the group algebra \(FG\) with defect group \(D\) and Cartan matrix \(C_B\). The author is interested in relations between \(R_B\), the set of eigenvalues of \(C_B\), and \(E_B\), the set of elementary divisors of \(C_B\). All eigenvalues of \(C_B\) are positive real numbers, and there is a unique largest one, denoted by \(\rho(B)\). The author conjectures that the following conditions are equivalent: (a) \(\rho(B)\in\mathbb{Z}\); (b) \(\rho(B)=|D|\); (c) \(R_B=E_B\); (d) \(R_B\subseteq\mathbb{Z}\). In the present paper, he considers the related question whether condition (d) implies the existence of a matrix \(U_B\in\text{GL}(l,R)\) whose columns consist of eigenvectors of \(C_B\). (A positive answer would give that (d) implies (c).) The author shows that his question has indeed a positive answer whenever \(B\) is Morita equivalent to its Brauer correspondent in \(N_G(D)\). He also checks that the answer is positive when \(D\) is cyclic, dihedral, semidihedral or (generalized) quaternion, or when \(G\) is \(p\)-solvable, or when \(B\) is the principal 3-block and \(|D|=9\).
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 5144428
zbMATH Keywords: blocks
zbMATH Keywords: defect groups
zbMATH Keywords: Cartan matrices
zbMATH Keywords: Perron-Frobenius eigenvalues
zbMATH Keywords: \(p\)-modular systems
zbMATH Keywords: group algebras
zbMATH Keywords: Brauer correspondents
reviewed by: Burkhard Külshammer
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.07.018
OpenAlex ID: W2010313971
cites work: Q3955560
cites work: Brauer correspondence for covering blocks of finite groups
cites work: Fixed points and \(\pi\)-characters in \(\pi\)-separable groups.
cites work: Rationality of eigenvalues of Cartan matrices in finite groups
cites work: Sone remarks on eigeyvalues of the cartan matrix in finite groups
cites work: A remark on blocks with dihedral defect groups in solvable groups
cites work: Broué's conjecture holds for principal 3-blocks with elementary Abelian defect group of order 9
cites work: The principal \(3\)-blocks of four- and five-dimensional projective special linear groups in non-defining characteristic
cites work: Morita Equivalent 3-Blocks of the 3-Dimensional Projective Special Linear Groups
cites work: Q3671358
cites work: Decomposition numbers of \(\text{Sp}(4,q)\)
cites work: Q4165490
cites work: Cellular algebras and Cartan matrices.
cites work: Eigenvalues and elementary divisors of Cartan matrices of cyclic blocks with \(l(B)\leqslant 5\) and tame blocks.
DOI: 10.1016/J.JALGEBRA.2006.07.018
description: scientific article; zbMATH DE number 5144428