Bijective proofs of some coinversion identities related to Macdonald polynomials

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.

10 pages

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