Journal article
A Shuffle Theorem for Paths Under Any Line
FORUM OF MATHEMATICS PI, v 11, e5
22 Feb 2023
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 Hall-Littlewood polynomials.
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Details
- Title
- A Shuffle Theorem for Paths Under Any Line
- Publication Details
- FORUM OF MATHEMATICS PI, v 11, e5
- Publisher
- CAMBRIDGE UNIV PRESS; CAMBRIDGE
- Grant note
- JB was supported by NSF grant DMS-1855784 and 2154282. JM was supported by Simons Foundation grant 821999 and NSF grant DMS-2154281. JM and GS were supported by NSF grant DMS-1855804.
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Drexel University
- Web of Science ID
- WOS:000937005000001
- Scopus ID
- 2-s2.0-85148995546
- Other Identifier
- 991021861189604721
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- Collaboration types
- Domestic collaboration
- Web of Science research areas
- Mathematics
- Mathematics, Applied