论文标题
通过模块化表示块进行分类II
Categorification via blocks of modular representations II
论文作者
论文摘要
Bernstein，Frenkel和Khovanov使用类别$ \ Mathcal {O} $ for $ \ Mathfrak {Slfrak {SL} _N $构建了$ \ Mathfrak {SL} _2 $标准表示的张量产品的分类。在较早的工作中，我们使用字段$ \ textbf {K textbf {k} $的特征性$ p> n $的$ \ mathfrak {sl} _n $的表示块构建了一个积极的特征模拟，带有零frobenius字符，零harish-chandra角色。在本文中，我们扩展了这些结果并构建了一个分类$ \ mathfrak {sl} _k $ - 遵循苏森的方法，通过考虑$ \ mathfrak {sl} _n $的模块化表示的更单一的模块化表示。我们认为零和非零frobenius中心特征。在前一种环境中，我们构建了这些分类的分级提升，相当于凯蒂斯，坎尼泽和licata的几何结构。我们在工作中构建的两个几何分类之间建立了二元性，并解决了他们的猜想。对于非零的Frobenius Central字符，我们表明，可以使用Cautis和Kamnitzer进行分类对称二元性的几何方法，可用于使用$ \ Mathfrak {SL} _N $的模块化表示的单数模块表示构造我们的分类升降。
Bernstein, Frenkel and Khovanov have constructed a categorification of tensor products of the standard representation of $\mathfrak{sl}_2$ using singular blocks of category $\mathcal{O}$ for $\mathfrak{sl}_n$. In earlier work, we construct a positive characteristic analogue using blocks of representations of $\mathfrak{sl}_n$ over a field $\textbf{k}$ of characteristic $p > n$, with zero Frobenius character, and singular Harish-Chandra character. In the present paper, we extend these results and construct a categorical $\mathfrak{sl}_k$-action, following Sussan's approach, by considering more singular blocks of modular representations of $\mathfrak{sl}_n$. We consider both zero and non-zero Frobenius central character. In the former setting, we construct a graded lift of these categorifications which are equivalent to a geometric construction of Cautis, Kamnitzer and Licata. We establish a Koszul duality between two geometric categorificatons constructed in their work, and resolve a conjecture of theirs. For non-zero Frobenius central characters, we show that the geometric approach to categorical symmetric Howe duality by Cautis and Kamnitzer can be used to construct a graded lift of our categorification using singular blocks of modular representations of $\mathfrak{sl}_n$.
