Symmetric and Nonsymmetric Macdonald Polynomials via a Path Model with a Pseudo-crystal Structure
Quantum Algebra
2022-10-27 v1 Combinatorics
Representation Theory
Abstract
In this paper we derive a counterpart of the well-known Ram-Yip formula for symmetric and nonsymmetric Macdonald polynomials of arbitrary type. Our new formula is in terms of a generalization of the Lakshmibai-Seshadri paths (originating in standard monomial theory), which we call pseudo-quantum Lakshmibai-Seshadri (LS) paths. This model carries less information than the alcove walks in the Ram-Yip formula, and it is therefore more efficient. Furthermore, we construct a connected pseudo-crystal structure on the pseudo-quantum LS paths, which is expected to lead to a simple Littlewood-Richardson rule for multiplying Macdonald polynomials. By contrast with the Kashiwara crystals, our pseudo-crystals have edges labeled by arbitrary roots.
Keywords
Cite
@article{arxiv.2210.14464,
title = {Symmetric and Nonsymmetric Macdonald Polynomials via a Path Model with a Pseudo-crystal Structure},
author = {Cristian Lenart and Satoshi Naito and Fumihiko Nomoto and Daisuke Sagaki},
journal= {arXiv preprint arXiv:2210.14464},
year = {2022}
}