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

2023/03/28 by Nicolle González, González, Nicolle, José Simental +3
Mathematics · Physics and Astronomy · #05E05 #05E10 (Primary) #05E14 (Secondary) #20C08 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2303.15694

openalex publication_date 2023/03/28 · openalex created_date 2023/03/31 · openalex updated_date 2026/07/28

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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