A Shuffle Theorem for Paths Under Any Line
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 $GL_{l}$ characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for nonsymmetric HallLittlewood polynomials.
 Publication:

arXiv eprints
 Pub Date:
 February 2021
 DOI:
 10.48550/arXiv.2102.07931
 arXiv:
 arXiv:2102.07931
 Bibcode:
 2021arXiv210207931B
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Quantum Algebra;
 Mathematics  Representation Theory;
 Primary: 05E05;
 Secondary: 16T30
 EPrint:
 43 pages, 7 figures