Categorification via blocks of modular representations for \(\mathfrak{sl}_n\). (English) Zbl 1521.17037
Summary: 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 Grassmannians. 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 Grassmannians. This is part of a larger project to give a combinatorial approach to Lusztig’s conjectures for representations of Lie algebras in positive characteristic.
MSC:
| 17B50 | Modular Lie (super)algebras |
| 22E47 | Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 14L35 | Classical groups (algebro-geometric aspects) |
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