A Shuffle Theorem for Paths Under Any Line

Abstract

AbstractWe 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 whosexandyintercepts 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.