Catalan numbers and noncommutative Hilbert schemes

Valery Lunts (Department of Mathematics, Indiana University, Bloomington, In., U.S.A.; and National Research University Higher School of Economics, Moscow, Russia)

Špela Špenko (Département de Mathématique, Université Libre de Bruxelles, Belgium)

Michel Van Den Bergh (Vakgroep Wiskunde, Universiteit Hasselt, Diepenbeek, Belgium; and Department of Mathematics & Data Science, Vrije Universiteit Brussels, Belgium)

Abstract

We find an explicit $S_n$-equivariant bijection between the integral points in a certain zonotope in $\mathbb{R}^n$, combinatorially equivalent to the permutahedron, and the set of m-parking functions of length n. This bijection restricts to a bijection between the regular $S_n$-orbits and $(m, n)$-Dyck paths, the number of which is given by the Fuss–Catalan number $A_n (m, 1)$. Our motivation came from studying tilting bundles on noncommutative Hilbert schemes. As a side result we use these tilting bundles to construct a semi-orthogonal decomposition of the derived category of noncommutative Hilbert schemes.

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The first author was supported by the Basic Research Program of the National Research University Higher School of Economics.

The second author is supported by a MIS grant from the National Fund for Scientific Research (FNRS) and an ACR grant from the Université Libre de Bruxelles.

The third author is a senior researcher at the Research Foundation Flanders (FWO). While working on this project he was supported by the ERC grant SCHEMES and the FWO grant G0D8616N: “Hochschild cohomology and deformation theory of triangulated categories”.

Received 2 July 2022

Received revised 4 January 2023

Accepted 11 February 2023

Published 15 May 2024