Hecke algebras, finite general linear groups, and Heisenberg categorification

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. Erratum: Due to a technical error, many lines in the diagrams were printed without indication of the corresponding direction (i.e. as simple lines instead of arrows). A corrected electronic version of the article can be downloaded here.