Counting characters in blocks. I
type: Item
instance of: scholarly article
title: Counting characters in blocks. I
zbMATH Open document ID: 0738.20011
DOI: 10.1007/BF01232023
author: Everett C. Dade
published in: Inventiones Mathematicae
publication date: 1992-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/144017
review text: The author presents a formula expressing the number of ordinary irreducible characters of a given height in a \(p\)-block \(B\) of a finite group \(G\) in terms of the numbers of ordinary irreducible characters of related heights in related blocks of certain \(p\)-local subgroups of \(G\). He shows by examples that this formula can fail if \(O_ p(G)\neq 1\) or if \(B\) has defect zero. Nevertheless, he conjectures that it holds in all other cases. Using results of \textit{R. Knörr} and \textit{G. Robinson} [J. Lond. Math. Soc., II. Ser. 39, 48-60 (1989; Zbl 0672.20005)], he shows that this conjecture would imply the Alperin weight conjecture. He verifies the conjecture for blocks with cyclic defect groups, and for all blocks of either the Mathieu group \(M_{11}\) or the Janko group \(J_ 1\). These easy verifications may actually be part of the eventual proof of a stronger version of the conjecture. The author claims that one such strong conjecture holds in general if it holds for each finite simple group \(G\). He also says that that strong conjecture coincides with the present one when \(G\) has trivial outer automorphism group and trivial Schur multiplier. Both \(M_{11}\) and \(J_ 1\) are among the eleven sporadic simple groups satisfying these two conditions. He hopes that someone will test the conjecture for the other nine, probably with computers for the largest ones.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C34
zbMATH DE Number: 4216119
zbMATH Keywords: numbers of ordinary irreducible characters
zbMATH Keywords: heights
zbMATH Keywords: \(p\)-blocks
zbMATH Keywords: finite groups
zbMATH Keywords: \(p\)-local subgroups
zbMATH Keywords: Alperin weight conjecture
zbMATH Keywords: Mathieu group \(M_{11}\)
zbMATH Keywords: Janko group \(J_ 1\)
zbMATH Keywords: sporadic simple groups
zbMATH Keywords: cyclic defect groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4216119