A major-index preserving map on fillings

TL;DR

This paper introduces a generalized major-index preserving map on fillings that connects combinatorial models for key and Macdonald polynomials, leading to new insights and solutions in algebraic combinatorics.

Contribution

It generalizes Mason's map to account for the major index and extends similar maps to models for modified Macdonald polynomials, addressing open questions.

Findings

Established a major-index preserving map for key polynomial models.

Extended the map to models for modified Macdonald polynomials at t=0.

Provided a combinatorial framework that solves conjectures and questions in algebraic combinatorics.

Abstract

We generalize a map by S. Mason regarding two combinatorial models for key polynomials, in a way that accounts for the major index. We also define similar variants of this map, that regards alternative models for the modified Macdonald polynomials at $t=0$, thus partially answer a question by J. Haglund. These maps imply certain uniqueness property regarding inversion-- and coinversion-free fillings, which allows us to generalize the notion of charge to a non-symmetric setting, thus answering a question by A. Lascoux. The analogous question in the symmetric setting proves a conjecture by K. Nelson.

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