Séminaire Lotharingien de Combinatoire, 85B.54 (2021), 12 pp.

Pavel Galashin and Thomas Lam

Positroids, Knots, and q,t-Catalan Numbers

Abstract. We relate the mixed Hodge structure on the cohomology of open positroid varieties (in particular, their Betti numbers over C and point counts over Fq) to Khovanov-Rozansky homology of the associated links. We deduce that the mixed Hodge polynomials of top-dimensional open positroid varieties are given by rational q,t-Catalan numbers. Via the curious Lefschetz property, this implies the q,t-symmetry and unimodality properties of rational q,t-Catalan numbers. We show that the q,t-symmetry phenomenon is a manifestation of Koszul duality for category O, and discuss relations with equivariant derived categories of flag varieties, and open Richardson varieties.

Received: December 1, 2020. Accepted: March 1, 2021. Final version: April 29, 2021.

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