Journal article

Categorification via blocks of modular representations for sl(n)

Vinoth Nandakumar, Gufang Zhao

CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES | CAMBRIDGE UNIV PRESS | Published : 2021

Abstract

Bernstein, Frenkel, and Khovanov have constructed a categorification of tensor products of the standard representation of, where they use singular blocks of category for and translation functors. Here we construct a positive characteristic analogue using blocks of representations of over a field 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 ..

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University of Melbourne Researchers

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Funding Acknowledgements

We are very much indebted to Roman Bezrukavnikov, who first suggested to us that the categorification [Oc] admits an abelian refinement using exotic t-structures and linear Koszul duality. We would like to thank Joel Kamnitzer, Jonathan Brundan, Sabin Cautis, Alexander Kleshchev, Simon Riche, Anthony Licata, Raphael Rouquier, Mikhail Khovanov, Will Hardesty, Michael Ehrig, Oded Yacobi, David Yang, You Qi and Kevin Coulembier for helpful discussions. We are grateful to Jim Humphreys for helpful comments on a previous version of this manuscript. ~e first author would also like to thank the University of Utah (in particular, Peter Trapa), and the University of Sydney (in particular, Gus Lehrer and Ruibin Zhang) for supporting this research. During the revision of this paper, the second named author was affiliated with the Institute of Science and Technology Austria, Hausel Group, supported by the Advanced Grant Arithmetic and Physics of Higgs moduli spaces No. coy Ac of the European Research Council.