# Horn inequalities for nonzero Kronecker coefficients

## TL;DR

This paper extends Horn inequalities, originally for Littlewood-Richardson coefficients, to the setting of nonzero Kronecker coefficients, providing new linear inequalities for these tensor product multiplicities.

## Contribution

It introduces a set of Horn inequalities applicable to triples of partitions with nonzero Kronecker coefficients, expanding the classical theory.

## Key findings

- Extended Horn inequalities to Kronecker coefficients.
- Established linear inequalities for nonzero Kronecker coefficients.
- Bridged the gap between Littlewood-Richardson and Kronecker coefficient theories.

## Abstract

The Kronecker coefficients and the Littlewood-Richardson coefficients are nonnegative integers depending on three partitions. By definition, these coefficients are the multiplicities of the tensor product decomposition of two irreducible representations of symmetric groups (resp. linear groups). By a classical Littlewood-Murnaghan's result the Kronecker coefficients extend the Littlewood-Richardson ones.The nonvanishing of a Littlewood-Richardson coefficient implies linear inequalities on the triple of partitions, called Horn inequalities. In thispaper, we extend the essential Horn inequalities to the triples of partitions corresponding to a nonzero Kronecker coefficient.