Jon Brundan 'The Heisenberg Category' (Part 1)

The Heisenberg category I and II

In these two talks, I will explain some foundational results about the (degenerate) Heisenberg monoidal category. This category was introduced 10 years ago by M. Khovanov (1009.3295) and encodes relations between natural transformations of the induction and restriction functors on representations of the symmetric group. It is perhaps the simplest of a number of “affine” diagrammatic monoidal categories of recent interest in representation theory, so it makes a great example pedagogically! In the first talk, I’ll try to explain the computation of its Grothendieck ring, which is a certain Z-form for the infinite-dimensional Heisenberg Lie algebra. In the second I’ll discuss the rich structure theory of Abelian module categories over the Heisenberg monoidal category. This includes the classical example of representations of symmetric groups, and many other categories from “type A” representation theory. The talks are based on two papers with A. Savage and B. Webster (1812.03255 and 1907.11988).