A survey of Heisenberg categorification via graphical calculus

TL;DR

This paper surveys different graphical categorifications of the Heisenberg algebra and Fock space, covering weak, geometrized, and strong approaches with diagrammatic and geometric methods.

Contribution

It provides a comprehensive overview of the development and variety of graphical categorification techniques for the Heisenberg algebra.

Findings

Comparison of weak and strong categorification methods

Overview of geometric approaches via Hilbert schemes

Discussion of planar diagrammatics in categorification

Abstract

In this expository paper we present an overview of various graphical categorifications of the Heisenberg algebra and its Fock space representation. We begin with a discussion of "weak" categorifications via modules for Hecke algebras and "geometrizations" in terms of the cohomology of the Hilbert scheme. We then turn our attention to more recent "strong" categorifications involving planar diagrammatics and derived categories of coherent sheaves on Hilbert schemes.