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

TL;DR

This paper introduces a new, efficient path model called pseudo-quantum LS paths for symmetric and nonsymmetric Macdonald polynomials, along with a pseudo-crystal structure that may simplify multiplication rules.

Contribution

It generalizes the Ram-Yip formula using pseudo-quantum LS paths and constructs a pseudo-crystal structure with edges labeled by arbitrary roots.

Findings

New pseudo-quantum LS path model for Macdonald polynomials

Efficient formula compared to alcove walk models

Pseudo-crystal structure with potential for simplified multiplication rules

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.