A bijective proof of a factorization formula for Macdonald polynomials at roots of unity

TL;DR

This paper provides a combinatorial bijective proof for a specific factorization formula of modified Macdonald polynomials at roots of unity, focusing on partitions with two columns, using Haglund, Haiman, and Loehr's interpretation.

Contribution

It introduces a new combinatorial bijective proof for the factorization formula of Macdonald polynomials at roots of unity for two-column partitions.

Findings

Proof applies to partitions with two columns

Utilizes combinatorial interpretation of Macdonald polynomials

Establishes a bijective proof for the factorization formula

Abstract

We give a combinatorial proof of the factorization formula of modified Macdonald polynomials when the parameter t is specialized at a primitive root of unity. Our proof is restricted to the special case of partitions with 2 columns. We mainly use the combinatorial interpretation of Haglund, Haiman and Loehr giving the expansion of the modified Macdonald polynomials on the monomial basis.