Higher rank (𝑞,𝑡)-Catalan polynomials, affine Springer fibers, and a finite rational shuffle theorem

González, Nicolle; Simental, José; Vazirani, Monica

doi:10.1090/tran/9115

Higher rank $(q,t)$-Catalan polynomials, affine Springer fibers, and a finite rational shuffle theorem
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by Nicolle González, José Simental and Monica Vazirani;

Trans. Amer. Math. Soc. 377 (2024), 5087-5127

DOI: https://doi.org/10.1090/tran/9115

Published electronically: May 17, 2024

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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 $\mathtt {dinv}$ statistic on rank $r$ semistandard $(m,n)$-parking functions and prove $\mathtt {codinv}$ 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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