Combinatorics on a family of reduced Kronecker coefficients

TL;DR

This paper computes the generating function for a specific family of reduced Kronecker coefficients, links it to plane partitions, and verifies the saturation conjecture, revealing their weakly increasing nature and quasipolynomial description.

Contribution

It provides the first explicit generating function for a family of reduced Kronecker coefficients and establishes their properties related to saturation and quasipolynomial behavior.

Findings

Family satisfies the saturation conjecture.

Coefficients are weakly increasing.

Family described by a quasipolynomial.

Abstract

The reduced Kronecker coefficients are particular instances of Kronecker coefficients that contain enough information to recover them. In this notes we compute the generating function of a family of reduced Kronecker coefficients. We also gives its connection to the plane partitions, which allows us to check that this family satisfies the saturation conjecture for reduced Kronecker coefficients, and that they are weakly increasing. Thanks to its generating function we can describe our family by a quasipolynomial, specifying its degree and period.