A survey of Heisenberg categorification via graphical calculus

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.

23 pages; v2: Some typos corrected and other minor improvements made; v3: Some small errors corrected; v4: Code corrected to fix problem with missing arrows on some diagrams