Tensor product of the Fock representation with its dual and the Deligne category

TL;DR

This paper investigates the structure of the tensor product of the basic Fock representation of sl(∞) with its shifted dual, using categorification via Deligne categories, and computes dimensions of key objects.

Contribution

It provides a detailed description of the tensor product structure and categorification of the Fock representation within Deligne categories, including dimension calculations.

Findings

Tensor product has a unique decreasing filtration with simple quotients.

Categorification via translation functors in Deligne categories is established.

Dimensions of standard and tilting objects are computed.

Abstract

We describe the structure of the tensor product of the basic Fock representation of sl(\infty) with its shifted dual. More precisely we prove that this tensor product has a unique decreasing filtration with simple quotients. We use the categorification of this representation via translation functors in the abelain envelope of the Deligne category GL(t) for integral t. We also compute dimensions of standard and tilting objects in this abelian envelope.