Polynomial functors and categorifications of Fock space II

TL;DR

This paper advances the categorification of Fock space representations using polynomial functors, connecting Heisenberg and affine Lie algebra actions, and exploring Schur-Weyl duality within higher representation theory.

Contribution

It introduces new categorifications of Fock space representations for Heisenberg and affine Lie algebras via polynomial functors, and links these to Schur-Weyl duality.

Findings

Categorifies Fock space of the Heisenberg algebra using polynomial functors.

Constructs commuting actions of affine Lie algebra and Heisenberg algebra on derived categories.

Shows Schur-Weyl duality as a morphism of categorification structures.

Abstract

We categorify various Fock space representations on the algebra of symmetric functions via the category of polynomial functors. In a prequel, we used polynomial functors to categorify the Fock space representations of type A affine Lie algebras. In the current work we continue the study of polynomial functors from the point of view of higher representation theory. Firstly, we categorify the Fock space representation of the Heisenberg algebra on the category of polynomial functors. Secondly, we construct commuting actions of the affine Lie algebra and the level p action of the Heisenberg algebra on the derived category of polynomial functors over a field of characteristic p > 0, thus weakly categorifying the Fock space representation of $\hat{gl}_p$. Moreover, we study the relationship between these categorifications and Schur-Weyl duality. The duality is formulated as a functor from the category of polynomial functors to the category of linear species. The category of linear species is known to carry actions of the Kac-Moody algebra and the Heisenberg algebra. We prove that Schur-Weyl duality is a morphism of these categorification structures.