Séminaire Lotharingien de Combinatoire, 89B.85 (2023), 12 pp.
Jonah Blasiak, Mark Haiman, Jennifer Morse, Anna Pun and George H. Seelinger
A Catalanimal Formula for Macdonald Polynomials
Abstract.
Catalanimals are rational functions encoding virtual GLl
character series. When truncated to the polynomial characters, they
have been shown to recover many important symmetric function
quantities with coefficients in Q(q,t) that arise from the
study of diagonal harmonics, including
∇mek and, more generally, ∇msλ. Providing
Catalanimal representatives of these quantities gave the necessary
tools to show ∇msλ is essentially a positive q,t-weighted sum of distinguished LLT
polynomials, thereby resolving the Loehr-Warrington
conjecture. Missing from this story was a Catalanimal description of
the modified Macdonald polynomials H~μ, which are
intimately linked to the ∇ operator. In
this abstract, we give a Catalanimal style expression for the modified
Macdonald polynomials and provide a positivity conjecture on the
entire GLl character series.
Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.
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