The nonsymmetric shuffle theorem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915520193757184 |
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| author | Blasiak, Jonah Haiman, Mark Morse, Jennifer Pun, Anna Seelinger, George H. |
| author_facet | Blasiak, Jonah Haiman, Mark Morse, Jennifer Pun, Anna Seelinger, George H. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_24040 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The nonsymmetric shuffle theorem Blasiak, Jonah Haiman, Mark Morse, Jennifer Pun, Anna Seelinger, George H. Combinatorics 05E05 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. |
| title | The nonsymmetric shuffle theorem |
| topic | Combinatorics 05E05 |
| url | https://arxiv.org/abs/2509.24040 |