Rickard equivalence
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| Rickard equivalence | |
|---|---|
| Name | Rickard equivalence |
| Field | Representation theory |
| Introduced | 1989 |
| Named after | Jeremy Rickard |
Rickard equivalence is an equivalence relation between block algebras of group algebras introduced in the context of modular representation theory. It refines notions arising in the work of Alperin, Brauer, and Burnside and connects to conjectures of Michel Broué and constructions used by Jeremy Rickard. The concept plays a central role in understanding how blocks associated to finite groups such as symmetric groups, alternating groups, general linear groups, and dihedral groups can share deep structural features.
Introduction
Rickard equivalence was formulated against the background of developments by Richard Brauer, John Green, and Gabriel Puig in block theory and derived categories as developed by Alexandre Grothendieck, Jean-Louis Verdier, and Bernhard Keller. It arises when comparing block algebras of finite groups such as sporadic groups and simple groups via derived equivalences constructed from complexes of bimodules. Early influential examples stem from work on blocks of symmetric groups and local methods inspired by the Local-global principle as studied by Isaacs and Navarro.
Definition and basic properties
A Rickard equivalence between two block algebras of finite groups, for instance between a block of finite groups G and a block of H, is given by a derived equivalence induced by a complex of bimodules satisfying perfection and homological criteria developed by Jeremy Rickard and elaborated by Rickard. Such an equivalence preserves invariants central to the work of Brauer, Fong, and Reynolds, including defect groups studied by Puig and fusion systems as investigated by Broto, Puig, and Alperin. Rickard equivalence implies Morita equivalence in many contexts considered by Morita and fits into the landscape shaped by results of Erdmann and Kessar.
Construction via complexes and tilting complexes
Constructions of Rickard equivalences typically employ bounded complexes of projective bimodules, especially tilting complexes inspired by work of Happel and Rickard. One approach uses splendid tilting complexes compatible with Puig’s theory of source algebras and splendid equivalences as in the literature of Linckelmann and Broué. Techniques draw on derived category machinery from Verdier and Neeman and on homotopy categories exploited by Keller and Happel. These complexes bridge classical constructions by Green and block induction methods associated with Brauer correspondence and transfer maps appearing in the work of Alperin and Thompson.
Relation to derived equivalences and Broué's conjecture
Rickard equivalence is a specific instance of derived equivalence in the sense of Grothendieck and Verdier, tailored to block algebras of finite groups. It provides the framework for the celebrated conjecture of Broué predicting that blocks with abelian defect groups are Rickard equivalent to their Brauer correspondents in the normalizer of a defect group. This conjecture connects to work by Dade, Külshammer, Linckelmann, and computational evidence from cases studied by Chuang and Rouquier. Progress has been achieved for families including symmetric groups, general linear groups over finite fields, and certain sporadic simple groups via techniques of Rickard and Kessar.
Examples and classification results
Explicit Rickard equivalences have been constructed for blocks of symmetric groups and alternating groups in low defect by work of Chuang and Rouquier, and for blocks with cyclic defect groups following classical results by Dade and Brauer. Classification results for blocks with Klein four defect groups involve contributions by Erdmann and Kessar, while dihedral, semidihedral, and quaternion defect cases have been analyzed in studies by Brauer and Alperin. Computational verifications for small groups use techniques developed by GAP and Magma with implementations influenced by researchers such as Lux and Holt.
Applications in modular representation theory
Rickard equivalence underpins structural results in modular representation theory involving decomposition numbers and source algebras appearing in the work of Brauer, Green, and Alperin. It informs local-global conjectures studied by Isaacs and Navarro and is applied to block invariants relevant to classification programs for finite simple groups such as those involving Lie type groups (e.g. Chevalley groups, Steinberg constructions). Rickard equivalences are also instrumental in categorical approaches pursued by Chuang and Rouquier and intersect with geometric methods inspired by Deligne and Lusztig.