A Shuffle Theorem for Paths Under Any Line. (22nd February 2023)

• Record Type: Journal Article
• Authors: Blasiak, Jonah; Haiman, Mark; Morse, Jennifer; Pun, Anna; Seelinger, George H.
• Is Part Of: Forum of mathematics. Volume 11(2023)
• Publication Date: 2023-02-22
• Subjects: 05E05 -- 16T30; Mathematics -- Periodicals; 510
• DOI: 10.1017/fmp.2023.4
• Languages: English
• ISSNs: 2050-5086

Abstract

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.