Odd Grassmannian bimodules and derived equivalences for spin symmetric groups

TL;DR

This paper develops odd analogs of categorification results and proves Broué's Abelian Defect Conjecture for double covers of symmetric groups, introducing a new framework of odd Grassmannian bimodules and odd symmetric functions.

Contribution

It introduces odd Grassmannian bimodules and extends categorification techniques to odd settings, completing the proof of Broué's conjecture for certain spin groups.

Findings

Established odd analogs of sl(2)-categorification results.

Proved Broué's Abelian Defect Conjecture for double covers of symmetric groups.

Developed the theory of odd symmetric functions.

Abstract

We prove odd analogs of results of Chuang and Rouquier on sl(2)-categorification. Combined also with recent work of the second author with Livesey, this allows us to complete the proof of Brou\'e's Abelian Defect Conjecture for the double covers of symmetric groups. The article also develops the theory of odd symmetric functions initiated a decade ago by Ellis, Khovanov and Lauda. A key role in our approach is played by a 2-category consisting of odd Grassmannian bimodules over superalgebras which are odd analogs of equivariant cohomology algebras of Grassmannians. This is the odd analog of the category of Grassmannian bimodules which was at the heart of Lauda's independent approach to categorification of sl(2). We also construct an action of the odd Kac-Moody 2-category of sl(2) on the 2-category of odd Grassmannian bimodules, and use this to give a new proof of its non-degeneracy.