Hecke algebras, finite general linear groups, and Heisenberg categorification

TL;DR

This paper introduces a diagrammatic category that categorifies the infinite rank Heisenberg algebra and acts on modules of Hecke algebras and finite general linear groups, connecting algebraic and categorical structures.

Contribution

It constructs a new diagrammatic category as a q-deformation of Khovanov's, providing a categorification of the Heisenberg algebra and its action on module categories.

Findings

Categorification of the infinite rank Heisenberg algebra.

Natural action on modules for Hecke algebras and finite general linear groups.

Development of parabolic induction and restriction functors with biadjointness and cyclicity.

Abstract

We define a category of planar diagrams whose Grothendieck group contains an integral version of the infinite rank Heisenberg algebra, thus yielding a categorification of this algebra. Our category, which is a q-deformation of one defined by Khovanov, acts naturally on the categories of modules for Hecke algebras of type A and finite general linear groups. In this way, we obtain a categorification of the bosonic Fock space. We also develop the theory of parabolic induction and restriction functors for finite groups and prove general results on biadjointness and cyclicity in this setting.