Categorification via blocks of modular representations II

Abstract

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 $\text{k}$ 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 ${\mathfrak{sl}}_n$.

Too many typos, will be replaced with an updated version

References in corpus (7)

• 2-Kac-Moody algebras
• Localization of modules for a semisimple Lie algebra in prime characteristic
• Modular representation theory in type A via Soergel bimodules
• Rigidity in higher representation theory
• Categorification via blocks of modular representations for sl(n)
• Koszul duality and Frobenius structure for restricted enveloping algebras
• Modular representations in type A with a two-row nilpotent central character