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

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.