Categorical actions and multiplicities in the Deligne category $\underline{Rep}(GL_t)$

TL;DR

This paper explores the categorical actions of type A on Deligne categories, revealing new connections between Lie algebra actions, tensor products, and multiplicities in tilting objects, with distinctions based on whether the parameter t is integer or not.

Contribution

It introduces a detailed study of categorical type A actions on Deligne categories and their abelian envelopes, linking them to Lie algebra representations and tensor product structures.

Findings

Categorical actions categorify Lie algebra actions on Fock spaces.

For integer t, the category models a tensor product of categorical modules.

For non-integer t, the category is semisimple and models an exterior tensor product.

Abstract

We study the categorical type A action on the Deligne category $\mathcal{D}_t=\underline{Rep}(GL_t)$ (here $t \in \mathbb{C}$) and its "abelian envelope" $\mathcal{V}_t$ constructed in arXiv:1511.07699. For $t \in \mathbb{Z}$, this action categorifies an action of the Lie algebra $\mathfrak{sl}_{\mathbb{Z}}$ on the tensor product of the Fock space $\mathfrak{F}$ with $\mathfrak{F}_t^{\vee}$, its restricted dual "shifted" by $t$, as was suggested by I. Losev. In fact, this action makes the category $\mathcal{V}_t$ the tensor product (in the sense of Losev and Webster, arXiv:1303.1336) of categorical $\mathfrak{sl}_{\mathbb Z}$-modules $Pol$ and $Pol_t^{\vee}$. The latter categorify $\mathfrak{F}$ and $\mathfrak{F}_t^{\vee}$ respectively, the underlying category in both cases being the category of stable polynomial representations (also known as the category of Schur functors), as described by Hong and Yacobi, arXiv:1101.2456 (see also Losev, arXiv:1209.1067). When $t \notin \mathbb Z$, the Deligne category $\mathcal{D}_t$ is abelian semisimple, and the type A action induces a categorical action of $\mathfrak{sl}_{\mathbb{Z}} \times \mathfrak{sl}_{\mathbb{Z}}$. This action categorifies the $\mathfrak{sl}_{\mathbb{Z}} \times \mathfrak{sl}_{\mathbb{Z} }$-module $\mathfrak{F} \boxtimes \mathfrak{F}^{\vee}$, making $\mathcal{D}_t$ the exterior tensor product of the categorical $\mathfrak{sl}_{\mathbb Z}$-modules $Pol$, $Pol^{\vee}$. Along the way we establish a new relation between the Kazhdan-Lusztig coefficients and the multiplicities in the standard filtrations of tilting objects in $\mathcal{V}_t$.