Some inequalities between invariants of blocks
type: Item
instance of: scholarly article
title: Some inequalities between invariants of blocks
zbMATH Open document ID: 1011.20009
author: Tomoyuki Wada
author: Burkhard Külshammer
published in: Archiv der Mathematik
publication date: 2002-12-01T00:00:00Z
review text: The second author [Arch. Math. 73, No. 6, 407-413 (1999; Zbl 0959.20013)] showed if \(B\) is a \(p\)-block of a finite group \(G\) with Cartan matrix \(C=(c_{ij})\), \(k(B)\) ordinary irreducible characters, and \(\ell(B)\) irreducible Brauer characters, then \[ k(B)\leq\sum_{i=1}^{\ell(B)}c_{ii}-\sum_{i=1}^{\ell(B)-1}c_{i,i+1}. \] Here, the authors generalize this by showing that \[ k(B)\leq\sum_{1\leq i\leq j\leq\ell(B)}q_{ij}c_{ij},\text{ where }\sum_{1\leq i\leq j\leq\ell(B)}q_{ij}X_iX_j \] is any weakly positive integal quadratic form. Some conditions for equality to hold are also given.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 1837269
zbMATH Keywords: blocks
zbMATH Keywords: Brauer characters
zbMATH Keywords: Cartan matrices
zbMATH Keywords: integral quadratic forms
zbMATH Keywords: irreducible characters
reviewed by: William H. Gustafson
MaRDI profile type: Publication
full work available at URL: s00013-002-8288-z
OpenAlex ID: W2015870055
DOI: 10.1007/S00013-002-8288-Z
description: scientific article; zbMATH DE number 1837269