Catalan numbers and noncommutative Hilbert schemes

Abstract

We find an explicit S-n -equivariant bijection between the integral points in a certain zonotope in 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.