Combinatorics on several families of Kronecker coefficients related to plane partitions

TL;DR

This paper explores three families of Kronecker coefficients linked to plane partitions, providing combinatorial interpretations, verifying the saturation hypothesis, and analyzing their quasipolynomial properties.

Contribution

It introduces a new combinatorial framework connecting Kronecker coefficients with plane partitions and verifies the saturation hypothesis for these families.

Findings

Saturation hypothesis holds for the studied families

Established combinatorial interpretation via plane partitions

Determined degree and period of governing quasipolynomials

Abstract

We present a study of three families of Kronecker coefficients, which we describe in terms of reduced Kronecker coefficients. This study is grounded on the generating function of the coefficients, proved by a bijection between two combinatorial objects. This study includes the connection between plane partitions and these three families of reduced Kronecker coefficients, providing us their combinatorial interpretation. As an application, we verify that the saturation hypothesis holds for our three families of reduced Kronecker coefficients. The study also includes other interpretation in terms of the quasipolynomials that govern these families. We specify the degree and the period of these quasipolynomials. Finally, the direct relation between Kronecker coefficients and reduced Kronecker coefficients allows us to give some observations about the rate of growth of the Kronecker coefficients associated to the reduced Kronecker coefficients of the study.