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arXiv · 2210.14464

Symmetric and Nonsymmetric Macdonald Polynomials via a Path Model with a Pseudo-crystal Structure

Cristian Lenart·Satoshi Naito·Fumihiko Nomoto·Daisuke Sagaki
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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.

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Cristian Lenart, Satoshi Naito, Fumihiko Nomoto, Daisuke Sagaki. 2022-10-26. Symmetric and Nonsymmetric Macdonald Polynomials via a Path Model with a Pseudo-crystal Structure. https://arxiv.org/abs/2210.14464

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