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RoCK blocks of double covers of symmetric groups and generalized Schur algebras

Alexander Kleshchev

Abstract

We study blocks of the double covers of symmetric and alternating groups. The main result is a `local' description, up to Morita equivalence, of arbitrary defect RoCK blocks of these groups in terms of generalized Schur superalgebras corresponding to an explicit Brauer tree superalgebra. In view of the recent results on Broué's Conjecture for these groups, our result provides a `local' description of an arbitrary block of the double covers of symmetric and alternating groups up to derived equivalence.

RoCK blocks of double covers of symmetric groups and generalized Schur algebras

Abstract

We study blocks of the double covers of symmetric and alternating groups. The main result is a `local' description, up to Morita equivalence, of arbitrary defect RoCK blocks of these groups in terms of generalized Schur superalgebras corresponding to an explicit Brauer tree superalgebra. In view of the recent results on Broué's Conjecture for these groups, our result provides a `local' description of an arbitrary block of the double covers of symmetric and alternating groups up to derived equivalence.

Paper Structure

This paper contains 15 sections, 6 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Double covers and generalized Schur algebras
  3. Cyclotomic quiver Hecke superalgebras
  4. Preliminaries
  5. General conventions
  6. Partitions and compositions
  7. Symmetric group
  8. Graded superalgebras
  9. Graded superspaces and superalgebras
  10. Graded supermodules
  11. Graded Morita superequivalence
  12. Regrading
  13. Superalgebras and supermodules
  14. Brauer tree superalgebras and generalized Schur algebras
  15. Brauer tree superalgebras ${\mathsf{A}}_\ell$

Key Result

Lemma 3.4

Let $A$ be a graded superalgebra and $B$ a unital graded subsuperalgebra isomorphic to a graded matrix superalgebra. Then we have an isomorphism of graded superalgebras $B \otimes Z_{A}(B) \stackrel{{}_\sim}{\longrightarrow} A,\ b\otimes z \mapsto bz.$

Theorems & Definitions (19)

  • Example 3.3
  • Lemma 3.4
  • proof
  • Example 3.9
  • Example 3.10
  • Lemma 3.14
  • Lemma 3.18
  • proof
  • Lemma 3.19
  • proof
  • ...and 9 more