The derived categories of some blocks of symmetric groups and a conjecture of Broué
type: Item
instance of: scholarly article
title: The derived categories of some blocks of symmetric groups and a conjecture of Broué
zbMATH Open document ID: 0939.20009
author: Joseph Chuang
published in: Journal of Algebra
publication date: 1999-10-27T00:00:00Z
review text: Let \(p\) be a prime, let \(G\) be a finite group and let \(B\) be a \(p\)-block of \(G\) with defect group \(D\). Also let \(b\) be the \(p\)-block of \(N_G(D)\) that is the Brauer correspondent of \(B\). A well-known conjecture of M. Broué is: If \(D\) is Abelian, then the \(B\)-module and \(b\)-module categories are derived equivalent. A fundamental result of J. Rickard gives necessary and sufficient conditions for the module categories of two rings to be derived equivalent. For \(p\)-blocks \(B\) and \(B'\) of finite groups \(G\) and \(H\) resp., the (left) module categories of \(B\) and \(B'\) are derived equivalent if and only if there exists a bounded complex \(X\) in \(B'\text{-mod-}B\) such that (1) each term of \(X\) is projective in \(B'\text{-mod}\) and in \(\text{mod-}B\); (2) \(X^*\otimes_{B'}X\) is a split complex in \(B\text{-mod-}B\) with homology concentrated in degree \(0\) isomorphic to \(B\); and (3) \(X\otimes_{B}X^*\) is a split complex in \(B'\text{-mod-}B'\) with homology concentrated in degree \(0\) isomorphic to \(B'\). Such a complex is called a Rickard tilting complex. If, in addition, \(G\) and \(H\) have a common \(p\)-subgroup \(D\) which is a defect group of \(B\) and \(B'\) and each term of \(X\) is relatively \(\Delta D\)-projective, then \(X\) is called a splendid Rickard tilting complex and \(B\) and \(B'\) are said to be splendid Rickard equivalent. Broué and Rickard have revised Broué's original conjecture: If \(D\) is Abelian, then the categories \(B'\text{-mod}\) and \(b\text{-mod}\) are splendid Rickard equivalent. The main result of this paper is: If \(p>2\) and \(B\) is a \(p\)-block of a symmetric group with defect group \(D\cong C_p\times C_p\), then \(B\) and its Brauer correspondent \(b\) are splendid Rickard equivalent. Section 2 of this paper presents a proof of an unpublished result of Rickard that all \(p\)-blocks of all symmetric groups with defect group \(D\cong C_p\times C_p\) are derived equivalent. Since A. Ma…
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 1340606
zbMATH Keywords: module categories
zbMATH Keywords: \(p\)-blocks
zbMATH Keywords: finite groups
zbMATH Keywords: derived equivalences
zbMATH Keywords: Rickard tilting complexes
zbMATH Keywords: splendid Rickard equivalences
zbMATH Keywords: symmetric groups
zbMATH Keywords: principal blocks
zbMATH Keywords: defect groups
zbMATH Keywords: Morita equivalences
Wikidata QID: Q123008921
reviewed by: Morton E. Harris
MaRDI profile type: Publication
full work available at URL: jabr.1998.7780
OpenAlex ID: W1966888854
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DOI: 10.1006/JABR.1998.7780
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description: scientific article; zbMATH DE number 1340606