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arXiv · 2012.09745

Positroids, knots, and $q,t$-Catalan numbers

Pavel Galashin·Thomas Lam
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Abstract

We relate the mixed Hodge structure on the cohomology of open positroid varieties (in particular, their Betti numbers over $\mathbb{C}$ and point counts over $\mathbb{F}_q$) to Khovanov--Rozansky homology of 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 of cluster varieties, 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 $\mathcal{O}$, and discuss relations with open Richardson varieties and extension groups of Verma modules.

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Pavel Galashin, Thomas Lam. 2020-12-17. Positroids, knots, and $q,t$-Catalan numbers. https://arxiv.org/abs/2012.09745

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