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

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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.

Original languageEnglish
Article number105735
JournalJournal of Combinatorial Theory. Series A
Volume196
DOIs
Publication statusPublished - May 2023
MoE publication typeA1 Journal article-refereed

Keywords

  • Boolean cumulants
  • Heisenberg category
  • Normalized character
  • Symmetric group

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