Bijective proofs of some coinversion identities related to Macdonald polynomials

TL;DR

This paper constructs explicit bijections to prove coinversion identities related to Macdonald polynomials, resolving an open problem and providing combinatorial insights into their structure.

Contribution

It provides explicit bijective proofs of coinversion identities associated with Macdonald polynomials, addressing an open problem in combinatorial representation theory.

Findings

Explicit bijections for coinversion identities

Validation of combinatorial formulas for Macdonald polynomials

Resolution of an open problem in algebraic combinatorics

Abstract

This paper gives bijective proofs of some novel coinversion identities first discovered by Ayyer, Mandelshtam, and Martin (arxiv:2011.06117) as part of their proof of a new combinatorial formula for the modified Macdonald polynomials $\tilde{H}_{\mu}$. Those authors used intricate algebraic manipulations of $q$-binomial coefficients to prove these identities, which imply the existence of certain bijections needed in their proof that their formula satisfies the axioms characterizing $\tilde{H}_{\mu}$. They posed the open problem of constructing such bijections explicitly. We resolve that problem here.