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Dens, nests and the Loehr-Warrington conjecture
Journal article   Open access   Peer reviewed

Dens, nests and the Loehr-Warrington conjecture

J. Blasiak, M. Haiman, J. Morse, A. Pun and G. Seelinger
Journal of the American Mathematical Society
10 Jun 2025
DOI: https://doi.org/10.1090/jams/1057
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https://doi.org/10.1090/jams/1057View
Published, Version of Record (VoR) Open

Abstract

We prove and extend the longest-standing conjecture in ‘ q , t q,t -Catalan combinatorics,’ namely, the combinatorial formula for ∇ m s μ \nabla ^m s_{\mu } conjectured by Loehr and Warrington, where s μ s_{\mu } is a Schur function and ∇ \nabla is an eigenoperator on Macdonald polynomials. Our approach is to establish a stronger identity of infinite series of G L l GL_l characters involving Schur Catalanimals ; these were recently shown by the authors to represent Schur functions s μ [ − M X m , n ] s_{\mu }[-MX^{m,n}] in subalgebras Λ ( X m , n ) ⊂ E \Lambda (X^{m,n})\subset \mathcal {E} isomorphic to the algebra of symmetric functions Λ \Lambda over Q ( q , t ) \mathbb {Q} (q,t) , where E \mathcal {E} is the elliptic Hall algebra of Burban and Schiffmann. We establish a combinatorial formula for Schur Catalanimals as weighted sums of LLT polynomials, with terms indexed by configurations of nested lattice paths called nests , having endpoints and bounding constraints controlled by data called a den . The special case for Λ ( X m , 1 ) \Lambda (X^{m,1}) proves the Loehr-Warrington conjecture, giving ∇ m s μ \nabla ^m s_{\mu } as a weighted sum of LLT polynomials indexed by systems of nested Dyck paths. In general, for Λ ( X m , n ) \Lambda (X^{m,n}) our formula implies a new ( m , n ) (m,n) version of the Loehr-Warrington conjecture. In the case where each nest consists of a single lattice path, the nests in a den formula reduce to our previous shuffle theorem for paths under any line. Both this and the ( m , n ) (m,n) Loehr-Warrington formula generalize the ( k m , k n ) (km,kn) shuffle theorem proven by Carlsson and Mellit (for n = 1 n=1 ) and Mellit. Our formula here unifies these two generalizations.

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Collaboration types
Domestic collaboration
Web of Science research areas
Mathematics
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