Catalan numbers, parking functions, permutahedra and noncommutative Hilbert schemes

2022/07/25 by Lunts, Valery, Špenko, Špela, Bergh, Michel Van den
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2207.12046

Abstract

We find an explicit S_n-equivariant bijection between the integral points in a certain zonotope in ℝⁿ, 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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