Odd Grassmannian bimodules and derived equivalences for spin symmetric groups

2022/03/26 by Brundan, Jonathan, Kleshchev, Alexander · 4 citations
#17B10 #18D25 #20C30 #FOS: Mathematics #Group Theory (math.GR) #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2203.14149

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 Broué'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.

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• Isomeric Heisenberg and Kac-Moody categorification I
• A local \mathfrakgl_1|1-action on odd Khovanov homology
• On RoCK blocks of double covers of symmetric and alternating groups and the refined Broué conjecture
• Rickard's Derived Morita Theory: Review and Outlook

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