A Shuffle Theorem for Paths Under Any Line
A Shuffle Theorem for Paths Under Any Line
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DOI：
10.1017/fmp.2023.4
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发表时间：
2021-02
期刊：
影响因子：
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通讯作者：
J. Blasiak;M. Haiman;J. Morse;Anna Y. Pun;G. Seelinger
中科院分区：
文献类型：
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作者：
J. Blasiak;M. Haiman;J. Morse;Anna Y. Pun;G. Seelinger
摘要我们推广了Haglund等人的Shuffle定理及其$(Km，Kn)$形式。和Bergeron等人。并分别被Carlsson和Mellit以及Mellit证明。在我们的版本中，组合边的$(Km，kn)$Dyck路被位于其x和y截距不需要是整数的线段下的格路所代替，并且代数侧由Schiffmann代数算子公式或等价的显式提升算子公式给出。我们用LLT多项式的无穷级数形式表示的，将我们的组合恒等式导出为运算符的无穷级数的恒等式的多项式截断。所讨论的级数恒等式源于非对称Hall-Littlewood多项式的柯西恒等式。
Abstract We generalize the shuffle theorem and its $(km,kn)$ version, as conjectured by Haglund et al. and Bergeron et al. and proven by Carlsson and Mellit, and Mellit, respectively. In our version the $(km,kn)$ Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whose x and y intercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of $\operatorname {\mathrm {GL}}_{l}$ characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for nonsymmetric Hall–Littlewood polynomials.