Multiplicity-free key polynomials

TL;DR

This paper classifies which key polynomials are multiplicity-free by using combinatorial models, providing a clear criterion for multiplicity-freeness in the context of Demazure modules of type A.

Contribution

It introduces a classification of multiplicity-free key polynomials using two combinatorial models, offering a new criterion for multiplicity-freeness.

Findings

Identifies a sufficient condition for quasi-key polynomials to be multiplicity-free.

Uses combinatorial models by Kohnert and Assaf-Searles for the classification.

Provides a complete classification of multiplicity-free key polynomials.

Abstract

The key polynomials, defined by A. Lascoux-M.-P. Sch\"utzenberger, are characters for the Demazure modules of type A. We classify multiplicity-free key polynomials. The proof uses two combinatorial models for key polynomials. The first is due to A. Kohnert. The second is by S. Assaf-D. Searles, in terms of quasi-key polynomials. Our argument proves a sufficient condition for a quasi-key polynomial to be multiplicity-free.