More cancellation theorems for conjectures of Alperin and Dade
type: Item
instance of: scholarly article
title: More cancellation theorems for conjectures of Alperin and Dade
zbMATH Open document ID: 1005.20014
author: Geoffrey R. Robinson
published in: Journal of Algebra
publication date: 2003-02-19T00:00:00Z
review text: Let \(G\) be a finite group, \(p\) a prime and \(B\) a \(p\)-block of \(G\). Alperin's weight conjecture (AWC) asserts that \[ \ell(B)=\sum_{Q/G}f^{(B)}_0(N_G(Q)/Q), \] where \(\ell(B)\) denotes the number of irreducible Brauer characters in \(B\), \(Q/G\) is a set of representatives for the \(G\)-conjugacy classes of \(p\)-subgroups of \(G\) and \(f^{(B)}_0(N_G(Q)/Q)\) denotes the number of \(p\)-blocks of defect \(0\) of \(N_G(Q)/Q\) contained in the \(B\)-Brauer correspondent blocks of \(N_G(Q)\). This conjecture was reformulated by R. Knörr and the author of this paper. This condition, equivalent to (AWC), was subsequently reformulated by E. C. Dade, Dade's projective conjecture (DPC), which is as follows: Let \(Z\) be a central \(p\)-subgroup of \(G\) and let \(\lambda\) be a linear character of \(Z\). Suppose that the defect group of \(B\) strictly contains \(Z\). Then \[ \sum_{\sigma\in{\mathcal N}(G,Z)/G}(-1)^{|\sigma|}k_d(B_\sigma,\lambda)=0\tag{1} \] for any integer \(d\geq 0\) and any such \(\lambda\) where \({\mathcal N}(G,Z)/G\) denotes a set of representatives for the \(G\)-conjugacy classes of strictly increasing normal chains of \(p\)-subgroups of \(G\) of the form \(\sigma=(Z=Q_0<Q_1<\cdots<Q_n)\) where all \(Q_i\triangleleft Q_n\), \(H_\sigma\) is the \(G\)-stabilizer of \(\sigma\), \(B_\sigma\) denotes the sum of \(B\)-Brauer correspondent blocks of \(H_\sigma\) and \(k_d(B_\sigma,\lambda)\) denotes the number of complex irreducible characters of defect \(d\) in the blocks of \(B_\sigma\) that cover \(\lambda\). To verify DPC, it is no loss to assume that \(Z=O_p(G)\). Moreover in a previous work, the author has shown that (1) is equivalent to \[ \sum_{\sigma\in{\mathcal N}(G,Z)/G}(-1)^{|\sigma|}k_d(B_\sigma)=0\tag{2} \] for all integers \(d\geq 0\) where \(k_d(B_\sigma)\) denotes the number of complex irreducible characters of defect \(d\) in the blocks of \(B_\sigma\). Note that \[ \sum_{\sigma\in{\mathcal N}(G,Z)}(-1)^{|\sigma|}k_d(B_\sigma)=k_…
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 1750364
zbMATH Keywords: finite groups
zbMATH Keywords: Alperin's weight conjecture
zbMATH Keywords: numbers of irreducible Brauer characters
zbMATH Keywords: numbers of blocks
zbMATH Keywords: Dade's projective conjecture
zbMATH Keywords: linear characters
zbMATH Keywords: defect groups
zbMATH Keywords: chains of \(p\)-subgroups
zbMATH Keywords: numbers of complex irreducible characters
reviewed by: Morton E. Harris
MaRDI profile type: Publication
OpenAlex ID: W2084813545
Wikidata QID: Q123004149
cites work: Q3804896
cites work: Counting characters in blocks with cyclic defect groups. I
cites work: Q5577154
cites work: Brauer correspondence for covering blocks of finite groups
cites work: Some Remarks on a Conjecture of Alperin
cites work: Extensions of nilpotent blocks
cites work: Dade's projective conjecture for \(p\)-solvable groups
cites work: On a projective generalization of Alperin's conjecture
cites work: Cancellation theorems related to conjectures of Alperin and Dade
DOI: 10.1006/JABR.2001.8984
description: scientific article; zbMATH DE number 1750364