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arXiv · 2303.15694

Higher rank $(q,t)$-Catalan polynomials, affine Springer fibers, and a finite Rational Shuffle Theorem

Nicolle González·José Simental·Monica Vazirani
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Abstract

We introduce the higher rank $(q,t)$-Catalan polynomials and prove they equal truncations of the Hikita polynomial to a finite number of variables. Using affine compositions and a certain standardization map, we define a dinv statistic on rank $r$ semistandard $(m,n)$-parking functions and prove codinv counts the dimension of an affine space in an affine paving of a parabolic affine Springer fiber. Combining these results, we give a finite analogue of the Rational Shuffle Theorem in the context of double affine Hecke algebras. Lastly, we also give a Bizley-type formula for the higher rank Catalan numbers in the non-coprime case.

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Nicolle González, José Simental, Monica Vazirani. 2023-03-28. Higher rank $(q,t)$-Catalan polynomials, affine Springer fibers, and a finite Rational Shuffle Theorem. https://arxiv.org/abs/2303.15694

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