Generalization of the Gale-Ryser theorem

TL;DR

This paper establishes a new inequality for Kostka-Foulkes polynomials, providing lower bounds for Kostka numbers and offering a novel proof of a key weight-multiplicity criterion in representation theory.

Contribution

It introduces a generalized inequality for Kostka-Foulkes polynomials, advancing understanding of their properties and applications in algebraic combinatorics.

Findings

Derived a new inequality for Kostka-Foulkes polynomials

Provided lower bounds for Kostka numbers

Presented a new proof of the Berenstein-Zelevinsky criterion

Abstract

We prove an inequality for the Kostka-Foulkes polynomials $K_{\lambda ,\mu}(q)$. As a corollary, we obtain a nontrivial lower bound for the Kostka numbers and a new proof of the Berenstein-Zelevinsky weight-multiplicity-one-criterium.