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Categorification via blocks of modular representations for $\mathfrak {sl}_n$

Part of: Lie groups Special varieties Algebraic groups

Published online by Cambridge University Press:  19 May 2020

Vinoth Nandakumar  and
Gufang Zhao
Vinoth Nandakumar*
Affiliation:
Max Planck Institute of Mathematics (Bonn) Vivatsgasse 7, 53111Bonn, Germany and School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia
Gufang Zhao
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA01003, USA and The University of Melbourne, School of Mathematics and Statistics, Parkville, VIC3010, Australia e-mail: gufangz@unimelb.edu.au
*
e-mail: vinoth.nandakumar@sydney.edu.au
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Abstract

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Bernstein, Frenkel, and Khovanov have constructed a categorification of tensor products of the standard representation of $\mathfrak {sl}_2$ , where they use singular blocks of category $\mathcal {O}$ for $\mathfrak {sl}_n$ and translation functors. Here we construct a positive characteristic analogue using blocks of representations of $\mathfrak {s}\mathfrak {l}_n$ over a field $\mathbf {k}$ of characteristic p with zero Frobenius character, and singular Harish-Chandra character. We show that the aforementioned categorification admits a Koszul graded lift, which is equivalent to a geometric categorification constructed by Cautis, Kamnitzer, and Licata using coherent sheaves on cotangent bundles to Grassmanians. In particular, the latter admits an abelian refinement. With respect to this abelian refinement, the stratified Mukai flop induces a perverse equivalence on the derived categories for complementary Grassmanians. This is part of a larger project to give a combinatorial approach to Lusztig’s conjectures for representations of Lie algebras in positive characteristic.

Keywords

Categorificationmodular representationFourier–Mukai transformlocalization

MSC classification

Primary: 22E47: Representations of Lie and real algebraic groups
Secondary: 14M15: Grassmannians, Schubert varieties, flag manifolds 14L35: Classical groups (geometric aspects)
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 4 , August 2021 , pp. 1095 - 1123
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Canadian Mathematical Society 2020

Footnotes

Dedicated to our friend, Dmitry Vaintrob

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