Trace and categorical sl(n) representations

TL;DR

This paper explores the trace decategorification of a 2-category related to $ ext{sl}_n$ representations, explicitly computes the induced current algebra action on cohomology rings, and offers a new proof of local Weyl modules' character formula.

Contribution

It explicitly computes the current algebra action on cohomology rings via trace functor and connects it to local Weyl modules, providing a new proof of their character formula.

Findings

Explicit computation of $ ext{sl}_n[t]$ action on cohomology rings.

Identification of the module with local Weyl modules.

New proof of the character formula for local Weyl modules.

Abstract

Khovanov-Lauda define a 2-category $\mathcal{U}$ such that the split Grothendieck group $K_0(\mathcal{U})$ is isomorphic to an integral version of the quantized universal enveloping algebra $\mathbf{U}(\mathfrak{sl}_n)$, $n \geq 2$. Beliakova-Habiro-Lauda-Webster prove that the trace decategorification of the Khovanov-Lauda 2-category is isomorphic to the the current algebra $\mathbf{U}(\mathfrak{sl}_n [t])$ - the universal enveloping algebra of the Lie algebra $ \mathfrak{sl}_n \otimes \mathbb{C} [t]$. A 2-representation of $\,\mathcal{U}$ is a 2-functor from $\mathcal{U}$ to a linear, additive 2-category. In this note we are interested in the 2-representation, defined by Khovanov-Lauda using bimodules over cohomology rings of flag varieties. This 2-representation induces an action of the current algebra $\mathbf{U}(\mathfrak{sl}_n [t])$ on the cohomology rings. We explicitly compute the action of $\mathbf{U}(\mathfrak{sl}_n [t])$ generators using the trace functor. It turns out that the obtained current algebra module is related to another family of $\mathbf{U}(\mathfrak{sl}_n [t])$-modules, called local Weyl modules. Using known results about the cohomology rings, we are able to provide a new proof of the character formula for the local Weyl modules.