The gluing problem in the fusion systems of the symmetric, alternating and linear groups.
type: Item
instance of: scholarly article
title: The gluing problem in the fusion systems of the symmetric, alternating and linear groups.
zbMATH Open document ID: 1247.20009
author: Assaf Libman
published in: Journal of Algebra
publication date: 2012-02-11T00:00:00Z
full work available at URL: http://hdl.handle.net/2164/2703
review text: Let \(p\) be a prime, \(k\) be an algebraically closed field of characteristic \(p\), \(\mathcal F\) be a saturated fusion system on a \(p\)-group \(S\) and \(\mathcal F^c\) be the full subcategory of the \(\mathcal F\)-centric subgroups. We have a natural map \[ \gamma\colon H^i(\mathcal F^c;k^\times)\to\varprojlim_{[S(\mathcal F^c)]}A^i_{\mathcal F}, \] where \(A^i_{\mathcal F}\), \(i\geq 0\), is the functor which associates to an equivalence class \([P_\bullet]\) of simplices in \(\mathcal F^c\), the Abelian group \(H^i(\Aut_{\mathcal F}(P_\bullet);k^\times)\); \([S(\mathcal F^c)]\) denotes the poset of equivalence classes of simplices in \(\mathcal F^c\). We say that Linckelmann's gluing problem for an element \((\alpha_{[P_\bullet]})\) of \(\varprojlim_{[S(\mathcal F^c)]}A^i_{\mathcal F}\) has a solution if this element is in the image of \(\gamma\). The main result of this paper is that the gluing problem for any element in \(\varprojlim_{[S(\mathcal F^c)]}A^2_{\mathcal F}\) has a solution in the fusion systems of the symmetric groups and the alternating groups and also \(\text{GL}_d(p^r)\) and \(\text{SL}_d(p^r)\).
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 55N91
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 55N25
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 6005433
zbMATH Keywords: saturated fusion systems
zbMATH Keywords: Linckelmann gluing problem
zbMATH Keywords: symmetric groups
zbMATH Keywords: alternating groups
zbMATH Keywords: equivariant homotopy theory
zbMATH Keywords: centric subgroups
reviewed by: Saïd Zarati
MaRDI profile type: Publication
OpenAlex ID: W2101544066
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DOI: 10.1016/J.JALGEBRA.2011.06.007
description: scientific article; zbMATH DE number 6005433