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The nonsymmetric shuffle theorem
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The nonsymmetric shuffle theorem

Jonah Blasiak, Mark Haiman, Jennifer Morse, Anna Pun and George H Seelinger
ArXiv.org
28 Sep 2025
DOI: https://doi.org/10.48550/arxiv.2509.24040
url
https://arxiv.org/pdf/2509.24040View
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Abstract

Mathematics - Combinatorics
The shuffle conjecture of Haglund et al. expresses the symmetric function$\nabla e_n$as a sum over labeled Dyck paths. Here$\nabla$is an operator on symmetric functions defined in terms of its diagonal action on the basis of modified Macdonald polynomials. The shuffle conjecture was later refined by Haglund-Morse-Zabrocki to the compositional shuffle conjecture, expressing$\nabla C_α$as a sum over labeled Dyck paths with touchpoints specified by$α$ , where$C_α$is a compositional Hall-Littlewood polynomial. Carlsson-Mellit settled both versions by developing the theory of a variant of the DAHA called the double Dyck path algebra. In a recent paper, we discovered a notion of nonsymmetric plethsym which led us to a construction of modified nonsymmetric Macdonald polynomials$\mathsf{H}_{η|λ}(\mathbf{x};q,t)$ . These polynomials Weyl symmetrize to their symmetric counterparts and are conjecturally atom positive. Here we introduce a nonsymmetric version$\boldsymbol{\nabla}$of$\nabla$ , now acting diagonally on the basis given by the functions$\mathsf{H}_{η|λ}(\mathbf{x};q,t)$ . Weaving together our theory with results of Carlsson-Mellit and Mellit, we establish a nonsymmetric version of the compositional shuffle theorem, which equates$\boldsymbol{\nabla}^{-1}$applied to a nonsymmetric version$\mathsf{C}_α$of$C_α$with a sum over flagged labeled Dyck paths with touchpoints given by$α$ . This combinatorial sum is conjecturally atom positive, refining the known Schur positivity of its symmetric counterpart.

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