Polynomial functors and categorifications of Fock space II

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 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. The classical pictureWe begin by reviewing various structures related to the algebra of symmetric functions.

Quantum polynomial functors

“…We hope this paper is a significant step in this program. This work is inspired by our previous works [HY1,HY2,HTY] on polynoimal functors and categorifications.…”

A Categorification of the Boson–Fermion Correspondence via Representation Theory of sl(∞)

“…It has been further extended in [LS1] and a survey of related works can be found in [LS2]. Another categorification of the bosonic Fock space and the action of the Lie algebra sl(∞) has been obtained in a series of papers [HY1], [HTY], [HY2]. This approach is closest to ours, though our tensor representations of sl(∞) are "rational" (not "polynomial") which leads to a non-semisimple category.…”

Polynomial functors and categorifications of Fock space

“…In the sequel to this work we continue the study of P from the point of view of higher representation theory [HY2]. We show that Khovanov's category H naturally acts on P, and this gives a categorification of the Fock space representation of the Heisenberg algebra when char(k) = 0.…”