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.

中文翻译：

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任何线下路径的洗牌定理

我们推广洗牌定理及其 $（公里，千）$ 版本，正如 Haglund 等人推测的那样。和伯杰龙等人。并分别由 Carlsson 和 Mellit 以及 Mellit 证明。在我们的版本中 $（公里，千）$ 组合侧的 Dyck 路径被位于线段下方的格子路径替换，该线段X和是截距不必是整数，代数边由 Schiffmann 代数运算符公式或等效的显式提升运算符公式给出。我们将组合恒等式导出为无限级数恒等式的多项式截断 $\operatorname {\mathrm {GL}}_{l}$ 字符，以 LLT 多项式的无限系列版本表示。所讨论的级数恒等式来自非对称 Hall-Littlewood 多项式的柯西恒等式。