A reduction theorem for Dade's projective conjecture
type: Item
instance of: scholarly article
title: A reduction theorem for Dade's projective conjecture
zbMATH Open document ID: 1459.20005
DOI: 10.4171/JEMS/688
author: Britta Späth
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2017-04-18T00:00:00Z
review text: In this paper, the author presents a strategy to reduce Dade's conjectures to statements on quasisimple groups, by formulating the Character Triple Conjecture, which a strengthening of Dade's extended projective conjecture. Let \(p\) be a prime, \(d\) an integer, \(G\) a finite group with \(\mathrm{O}_p(G) \le Z(G)\), and \(B\) a \(p\)-block of \(G\) with a noncentral defect group. Denote by \(\mathfrak{P}(G|\mathrm{O}_p(G))\) the set of \(p\)-chains \(\mathbb{D}=(D_0= \mathrm{O}_p(G) \lneq D_1 \lneq \dots \lneq D_l)\) of \(G\) starting with \(\mathrm{O}_p(G)\). Here, \(l=|\mathbb{D}|\) is the length of \(\mathbb{D}\). The set of \(p\)-chains of even length is \(\mathfrak{P}(G|\mathrm{O}_p(G))_+\), and the set of \(p\)-chains of odd length is \(\mathfrak{P}(G|\mathrm{O}_p(G))_-\). Let \(G_\mathbb{D}\) be the normalizer of \(\mathbb{D}\) in \(G\), and \(B_\mathbb{D}\) the set of all \(p\)-blocks \(b\) of \(G_\mathbb{D}\) such that \(b^G = B\). For \(\chi \in \mathrm{Irr}(G)\), the defect \(\mathrm{d}(\chi)\) of \(\chi\) is defined by \(p^{\mathrm{d}(\chi)} \chi(1)_p = |G|_p\). For a set \(C\) of \(p\)-blocks of \(G\), the set \(\mathrm{Irr}^d(C)\) consists of all characters with defect \(d\) belonging to a block in \(C\). Denote by \(\mathcal{C}^d(B)_+\) the set of all pairs \((\mathbb{D},\theta)\) with \(\mathbb{D}\in \mathfrak{P}(G|\mathrm{O}_p(G))_+\) and \(\theta \in \mathrm{Irr}^d(B_\mathbb{D})\), and \(\mathcal{C}^d(B)_-\) is defined analogously. The group \(G\) acts on \(\mathcal{C}^d(B)_+\) and \(\mathcal{C}^d(B)_-\) by conjugation. For \((\mathbb{D},\theta)\in \mathcal{C}^d(B)_+\) denote by \(\overline{(\mathbb{D},\theta)}\) its \(G\)-orbit, and by \(\overline{\mathcal{C}^d(B)_+}\) and \(\overline{\mathcal{C}^d(B)_-}\) the corresponding sets of \(G\)-orbits. Dade's projective conjecture can be formulated as follows: There exists a bijection \(\Omega: \overline{\mathcal{C}^d(B)_+} \to \overline{\mathcal{C}^d(B)_-}\) such that \(\theta_{\mathrm{O}_p(G)}\) and …
reviewed by: Andrei Marcus
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20C34
zbMATH DE Number: 6705848
zbMATH Keywords: representations of finite groups
zbMATH Keywords: block theory
zbMATH Keywords: Dade's conjectures
Wikidata QID: Q123254471
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/jems/688
OpenAlex ID: W2644687854
description: scientific article; zbMATH DE number 6705848