Normalized characters of symmetric groups and Boolean cumulants via Khovanov's Heisenberg category

TL;DR

This paper explores the relationship between normalized characters of symmetric groups and Boolean cumulants of Young diagrams, using Khovanov's Heisenberg category to reveal polynomial structures and non-negative integer coefficients, deepening the link between representation theory and categorification.

Contribution

It demonstrates that normalized characters are polynomials of Boolean cumulants with non-negative integer coefficients, utilizing graphical categorification via Khovanov's Heisenberg category.

Findings

Normalized characters are polynomials of Boolean cumulants.

Coefficients in expansions are non-negative integers.

Graphical manipulations facilitate computations on Young diagrams.

Abstract

In this paper, we study relationships between the normalized characters of symmetric groups and the Boolean cumulants of Young diagrams. Specifically, we show that each normalized character is a polynomial of twisted Boolean cumulants with coefficients being non-negative integers, and conversely, that, when we expand a Boolean cumulant in terms of normalized characters, the coefficients are again non-negative integers. The main tool is Khovanov's Heisenberg category and the recently established connection of its center to the ring of functions on Young diagrams, which enables one to apply graphical manipulations to the computation of functions on Young diagrams. Therefore, this paper is an attempt to deepen the connection between the asymptotic representation theory and graphical categorification.