Categorification via blocks of modular representations II

TL;DR

This paper extends categorification of tensor products of rak{sl}_2 to rak{sl}_k using modular representations, establishing geometric dualities and constructing graded lifts in both zero and non-zero Frobenius character settings.

Contribution

It constructs a categorical rak{sl}_k-action via singular blocks of modular rak{sl}_n representations, generalizing previous rak{sl}_2 results and connecting to geometric categorifications.

Findings

Constructed a graded lift of categorifications matching geometric models.

Established Koszul duality between two geometric categorifications.

Resolved a conjecture by Cautis, Kamnitzer, and Licata.

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