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Abstract

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 $\mathfrak{sl}_n$ over a field $\textbf{k}$
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.