The Derived Categories of Some Blocks of Symmetric Groups and a Conjecture of Broué

✍ Scribed by Joseph Chuang

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 282 KB
• Volume: 217
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1998.7780

✦ Synopsis

A well-known conjecture of Broue in the representation theory of finite groups ínvolves equivalences of derived categories of blocks. The aim of this paper is to verify this conjecture for defect 2 blocks of symmetric groups. Actually we prove for these blocks a refinement of Broue's conjecture due to Rickard. ᮊ 1999 Ácademic Press 1. PRELIMINARIES Let p be a prime number and let O O be a complete discrete valuation ring with residue field k of characteristic p and fraction field K of characteristic zero, ''big enough'' for all the groups considered. We will use R to denote a ring which is either O O or k, and if V is an O O-module, we will usually write V in place of k m V. O O Let G be a finite group. By a block B of G we mean an indecomposable algebra summand of the group algebra O OG. Denote by kB the corresponding summand k m B of kG, and by KB the corresponding O O summand K m B of KG. The principal block of G will be denoted by O O Ž . B G . 0

Throughout this paper, all modules are finitely generated left modules, unless otherwise stated. Also if B and B X are blocks of finite groups, we will assume that for any RB X -RB-bimodule M the two R actions coincide. U Ž . X Denote by M the R-dual Hom M, R , which carries a natural RB-RB -R bimodule structure. b Ž . Let K B be the category whose objects are bounded complexes of B-modules and whose morphisms are chain maps modulo homotopy equiv-114