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Sahi, Stokman, and Venkateswaran have constructed, for each positive integer$n$, a family of Laurent polynomials depending on parameters $q$ and $k$ (inaddition 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 witharbitrary 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 Ramand Yip in order to give a combinatorial formula for the SSV polynomials interms of alcove walks. The formula immediately shows that the SSV polynomialssatisfy 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 basisfor the space of Laurent polynomials. The result is also used to show that theSSV polynomials have \emph{fewer} terms than the corresponding Macdonaldpolynomials. We also record an alcove walk formula for the naturalgeneralization of the permuted basement Macdonald polynomials. We thenconstruct a symmetrized variant of the SSV polynomials: these are symmetricwith respect to a conjugate of the Chinta-Gunnells Weyl group action and reduceto symmetric Macdonald polynomials when $n=1$. We obtain an alcove walk formulafor the symmetrized polynomials as well. Finally, we calculate the$q\rightarrow 0$ and $q\rightarrow \infty$ limits of the SSV polynomials andobserve that our combinatorial formula can be written in terms of alcove walkswith only positive and negative folds respectively. In both of these $q$-limitcases, we also observe a positivity result for the coefficients.

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