A remark on conjectures in modular representation theory
type: Item
instance of: scholarly article
title: A remark on conjectures in modular representation theory
zbMATH Open document ID: 0607.20005
DOI: 10.1007/BF01194096
published in: Archiv der Mathematik
publication date: 1987-01-01T00:00:00Z
review text: Let p be a prime, let B be a p-block of some finite group G with defect group D, and let \(k_ 0(B)\) denote the number of irreducible ordinary characters of height zero in B. In [Math. Z. 186, 41-47 (1984; Zbl 0534.20006)] \textit{J. Olsson} conjectured that \(k_ 0(B)\leq | D/D'|\) where D' denotes the commutator subgroup of D. In this note we show that this conjecture is a consequence of two other conjectures, Brauer's conjecture on the number k(B) of irreducible ordinary characters and the Alperin-McKay conjecture. Brauer's conjecture asserts that k(B)\(\leq | D|\), and the Alperin-McKay conjecture asserts that \(k_ 0(B)=k_ 0(b)\) where b denotes the Brauer correspondent of B in \(N_ G(D)\).
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 3981413
zbMATH Keywords: p-block
zbMATH Keywords: defect group
zbMATH Keywords: number of irreducible ordinary characters of height zero
zbMATH Keywords: Brauer's conjecture
zbMATH Keywords: Alperin-McKay conjecture
zbMATH Keywords: Brauer correspondent
author: Burkhard Külshammer
MaRDI profile type: Publication
Wikidata QID: Q123282395
cites work: Q3955560
cites work: On the k(GV) problem
cites work: Crossed products and blocks with normal defect groups
cites work: Character correspondence and p-blocks of P-solvable groups
cites work: On the number of characters in blocks of finite general linear, unitary and symmetric groups
cites work: Blocks and Normal Subgroups of Finite Groups
cites work: The number of blocks with a given defect group
full work available at URL: https://doi.org/10.1007/bf01194096
OpenAlex ID: W1974899783
description: scientific article; zbMATH DE number 3981413