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.

Bernstein、Frenkel 和 Khovanov 构建了 $\mathfrak {sl}_2$ 的标准表示的张量积的分类 ，其中他们使用类别 $\mathcal {O}$ 的 奇异块来 表示 $\mathfrak {sl}_n$ 和翻译函子。在这里，我们使用 $\mathfrak {s}\mathfrak {l}_n$ 在特征p的字段 $\mathbf {k}$ 上的表示块构建一个正特征模拟 具有零 Frobenius 字符和单数 Harish-Chandra 字符。我们表明，上述分类允许 Koszul 分级提升，这等效于由 Cautis、Kamnitzer 和 Licata 使用相切丛上的相干滑轮构造的几何分类到 Grassmanians。特别地，后者承认阿贝尔细化。关于这种阿贝尔细化，分层的 Mukai 触发器在互补 Grassmanians 的派生类别上引起了反常的等价。这是一个更大项目的一部分，该项目为 Lusztig 的猜想提供了一种组合方法，用于表示具有正特征的李代数。