2025/09/28 by Jonah Blasiak, Blasiak, Jonah, Mark Haiman +7 · 1 citation
Mathematics · #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2509.24040
The shuffle conjecture of Haglund et al. expresses the symmetric function ∇ en as a sum over labeled Dyck paths. Here ∇ 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 ∇ Cα as a sum over labeled Dyck paths with touchpoints specified by α, where Cα 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 Hη|λ(x;q,t). These polynomials Weyl symmetrize to their symmetric counterparts and are conjecturally atom positive. Here we introduce a nonsymmetric version \boldsymbol∇ of ∇, now acting diagonally on the basis given by the functions Hη|λ(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∇-1 applied to a nonsymmetric version Cα of Cα 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.