Abacus proofs of Schur expansions of Macdonald polynomials

Loehr, Nicholas

doi:10.1090/proc/16092

Abacus proofs of Schur expansions of Macdonald polynomials
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by Nicholas A. Loehr

Proc. Amer. Math. Soc. 151 (2023), 59-71

DOI: https://doi.org/10.1090/proc/16092

Published electronically: August 12, 2022

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Abstract:

A famous problem in symmetric function theory is to find combinatorial formulas for Schur expansions of the Macdonald polynomials $\tilde {H}_{\mu }$. One such formula, valid for $\mu$ satisfying $\mu _1\leq 3$ and $\mu _2\leq 2$, involves Yamanouchi words weighted by Haglund’s statistics $\operatorname {inv}_{\mu }$ and $\operatorname {maj}_{\mu }$. Previous proofs of this formula use the technical machinery of crystals and dual equivalence graphs. We give a new, elementary, and fully bijective proof of this formula based on the abacus model for antisymmetrized Macdonald polynomials. An extension to the Schur expansion of $s_{\nu }\tilde {H}_{\mu }$ is also provided.

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