An Involution on Semistandard Skyline Fillings

TL;DR

This paper introduces an involution on semistandard skyline fillings, generalizing classical involutions, and uses it to provide a combinatorial proof of the specialization of nonsymmetric Macdonald polynomials to key polynomials.

Contribution

It constructs a new involution on semistandard skyline fillings, extending the Bender--Knuth involution, and demonstrates its compatibility with Demazure operators.

Findings

The involution generalizes classical Bender--Knuth involution.

Semistandard skyline fillings are shown to be compatible with Demazure operators.

Provides a new combinatorial proof for the specialization of nonsymmetric Macdonald polynomials.

Abstract

Non-attacking skyline fillings were used by Haglund, Haiman and Loehr to establish a combinatorial formula for nonsymmetric Macdonald polynomials. Semistandard skyline fillings are non-attacking skyline fillings with both major index and coinversion number equal to zero, which serve as a combinatorial model for key polynomials. In this paper, we construct an involution on semistandard skyline fillings. This involution can be viewed as a vast generalization of the classical Bender--Knuth involution. As an application, we obtain that semistandard skyline fillings are compatible with the Demazure operators, offering a new combinatorial proof that nonsymmetric Macdonald polynomials specialize to key polynomials.