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Title: 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 $\operatorname {\mathrm {GL}}_{l}$ 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.  more » « less
Award ID(s):
2154281 1855804
NSF-PAR ID:
10410649
Author(s) / Creator(s):
; ; ; ;
Date Published:
Journal Name:
Forum of Mathematics, Pi
Volume:
11
ISSN:
2050-5086
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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