The nonsymmetric shuffle theorem

Abstract: 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 \(\nabla C_\alpha\) as a sum over labeled Dyck paths with touchpoints specified by \(\alpha\), where \(C_\alpha\) 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}_{\eta

\lambda}(\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}_{\eta

\lambda}(\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 \(\mathsf{C}\alpha\) of \(C\alpha\) with a sum over flagged labeled Dyck paths with touchpoints given by \(\alpha\). This combinatorial sum is conjecturally atom positive, refining the known Schur positivity of its symmetric counterpart.

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Date published: Tuesday, September 30, 2025