The Newton polytope of the Kronecker product

TL;DR

This paper investigates the Newton polytope of the Kronecker product of Schur functions, proving special cases of a conjecture related to its monomial expansion and using Horn inequalities to analyze positivity conditions.

Contribution

It establishes special cases of a conjecture on the Newton polytope of the Kronecker product and links Horn inequalities to positivity of Kronecker coefficients.

Findings

Proved special cases of the Monical--Tokcan--Yong conjecture.

Derived necessary conditions for Kronecker coefficient positivity.

Connected Horn inequalities with the structure of the Newton polytope.

Abstract

We study the Kronecker product of two Schur functions $s_\lambda\ast s_\mu$, defined as the image of the characteristic map of the product of two $S_n$ irreducible characters. We prove special cases of a conjecture of Monical--Tokcan--Yong that its monomial expansion has a saturated Newton polytope. Our proofs employ the Horn inequalities for positivity of Littlewood-Richardson coefficients and imply necessary conditions for the positivity of Kronecker coefficients.