A combinatorial formula for Sahi, Stokman, and Venkateswaran's generalization of Macdonald polynomials

TL;DR

This paper provides a combinatorial alcove walk formula for Sahi, Stokman, and Venkateswaran's generalized Macdonald polynomials, called SSV polynomials, revealing their structure, basis properties, and limits, with implications for symmetry and positivity.

Contribution

It introduces a combinatorial alcove walk formula for SSV polynomials, proving their basis property, triangularity, and fewer terms than Macdonald polynomials, and explores their symmetrized variants and limits.

Findings

SSV polynomials satisfy a triangularity property with respect to Bruhat order.

SSV polynomials form a basis for Laurent polynomials.

The number of terms in SSV polynomials is fewer than in Macdonald polynomials.

Abstract

Sahi, Stokman, and Venkateswaran have constructed, for each positive integer $n$, a family of Laurent polynomials depending on parameters $q$ and $k$ (in addition to $\lfloor n/2\rfloor$ "metaplectic parameters"), such that the $n=1$ case recovers the nonsymmetric Macdonald polynomials and the $q\rightarrow\infty$ limit yields metaplectic Iwahori-Whittaker functions with arbitrary Gauss sum parameters. In this paper, we study these new polynomials, which we call SSV polynomials, in the case of $GL_r$. We apply a result of Ram and Yip in order to give a combinatorial formula for the SSV polynomials in terms of alcove walks. The formula immediately shows that the SSV polynomials satisfy a triangularity property with respect to a version of the Bruhat order, which in turn gives an independent proof that the SSV polynomials are a basis for the space of Laurent polynomials. The result is also used to show that the SSV polynomials have \emph{fewer} terms than the corresponding Macdonald polynomials. We also record an alcove walk formula for the natural generalization of the permuted basement Macdonald polynomials. We then construct a symmetrized variant of the SSV polynomials: these are symmetric with respect to a conjugate of the Chinta-Gunnells Weyl group action and reduce to symmetric Macdonald polynomials when $n=1$. We obtain an alcove walk formula for the symmetrized polynomials as well. Finally, we calculate the $q\rightarrow 0$ and $q\rightarrow \infty$ limits of the SSV polynomials and observe that our combinatorial formula can be written in terms of alcove walks with only positive and negative folds respectively. In both of these $q$-limit cases, we also observe a positivity result for the coefficients.