Abstract
AbstractBernstein, 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.
Highlights
In [ ], Bernstein, Frenkel and Khovanov categorify the action of sl on the tensor product (C )⊗n using singular blocks of category O for sln
In [ ], Frenkel, Kirillov and Khovanov show that the classes of the simple objects in these representation categories match up with the dual canonical basis in (C )⊗n, specialized at q = . ese results can be used to give a combinatorial approach to the Kazhdan– Lusztig Conjectures in type A, and categorical techniques have since been used widely in representation theory
We extend this approach to representation categories of Lie algebras in positive characteristic
Summary
In [ ], Bernstein, Frenkel and Khovanov categorify the action of sl on the tensor product (C )⊗n using singular blocks of category O for sln. We first construct a modular analogue of this result, using blocks of representations of the Lie algebra g ∶= sln defined over an algebraically closed field k of characteristic p > n; see Sections . E second main result of the present paper is on a graded li of the categorification from eorem A, which is equivalent to one constructed by Cautis, Kamnitzer, and Licata [ ]; see Section . E graded li of the modular representation categories in question, denoted by Modfg,,μgr r(Ug), is called the Koszul grading, constructed by Riche [ ], using a localization equivalence that builds upon the framework developed by Bezrukavnikov, Mirkovic, and Rumynin in [ ]. F−n+ r+ and an octuple of natural transforms making it a strong sl -categorification This categorification is equivalent to the categorification constructed by Cautis, Kamnitzer, and Licata [ ]. In Section , we discuss some open problems and further directions
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