Kronecker coefficients for (dual) symmetric inverse semigroups

TL;DR

This work extends the notion of Kronecker coefficients to semigroup settings, focusing on the symmetric inverse semigroup $IS_n$, its dual $I_n^*$, and the subquotient-bijection semigroup $PI_n^*$. The authors derive an explicit formula for the IS_n case in terms of classical symmetric-group invariants $\mathbf{g}_{\cdot,\cdot}^{\cdot}$ and $\mathbf{c}_{\cdot,\cdot}^{\cdot}$, and provide a structured, combinatorial framework for the dual and subquotient cases via 0/1-matrix data and induction/restriction among symmetric groups. They connect these Kronecker-type problems to Schur–Weyl duality for partition algebras, establishing a stability phenomenon for restriction-type coefficients and showing that tensor products of cell modules over partition algebras need not have a cell filtration. Overall, the paper advances understanding of Kronecker-type phenomena beyond groups, linking semigroup representation theory with classical combinatorial invariants and diagram algebras, with implications for FI-modules and related areas.

Abstract

We study analogues of Kronecker coefficients for symmetric inverse semigroups, for dual symmetric inverse semigroups and for the inverse semigroups of bijections between subquotients of finite sets. In all cases we reduce the problem of determination of such coefficients to some group-theoretic and combinatorial problems. For symmetric inverse semigroups, we provide an explicit formula in terms of the classical Kronecker and Littlewood--Richardson coefficients for symmetric groups.

Theorems & Definitions (33)

• Theorem 1
• Lemma 2
• proof
• Corollary 3
• Example 4
• Proposition 5
• proof
• Example 6
• Proposition 7
• proof