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Catalan numbers, parking functions, permutahedra and noncommutative Hilbert schemes

Valery Lunts, Špela Špenko, Michel Van den Bergh

TL;DR

The paper establishes a new explicit $S_n$-equivariant bijection between lattice points in the zonotope $\Delta^{m,n}_\tau$ (for admissible $\tau$) and $(m,n)$-parking functions, revealing that regular $S_n$-orbits correspond to $(m,n)$-Dyck paths counted by the Fuss-Catalan number $A_n(m,1)$. It then connects these combinatorial structures to the geometry of the noncommutative Hilbert schemes $H_{m,n}$ by presenting a GIT description and constructing tilting bundles $\mathcal{T}_\tau$, enabling a semi-orthogonal decomposition of the derived category $D( H_{m,n})$. The results provide a geometric proof that the number of regular orbits equals $A_n(m,1)$ and establish a framework in which the combinatorial and representation-theoretic aspects of $(m,n)$-parking functions, Dyck paths, and Fuss-Catalan numbers are encoded in the derived category via tilting theory. This work bridges polyhedral combinatorics with noncommutative algebraic geometry, suggesting explicit SOD components connected to the Azumaya locus and trace rings.

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.

Catalan numbers, parking functions, permutahedra and noncommutative Hilbert schemes

TL;DR

The paper establishes a new explicit -equivariant bijection between lattice points in the zonotope (for admissible ) and -parking functions, revealing that regular -orbits correspond to -Dyck paths counted by the Fuss-Catalan number . It then connects these combinatorial structures to the geometry of the noncommutative Hilbert schemes by presenting a GIT description and constructing tilting bundles , enabling a semi-orthogonal decomposition of the derived category . The results provide a geometric proof that the number of regular orbits equals and establish a framework in which the combinatorial and representation-theoretic aspects of -parking functions, Dyck paths, and Fuss-Catalan numbers are encoded in the derived category via tilting theory. This work bridges polyhedral combinatorics with noncommutative algebraic geometry, suggesting explicit SOD components connected to the Azumaya locus and trace rings.

Abstract

We find an explicit -equivariant bijection between the integral points in a certain zonotope in , combinatorially equivalent to the permutahedron, and the set of -parking functions of length . This bijection restricts to a bijection between the regular -orbits and -Dyck paths, the number of which is given by the Fuss-Catalan number . 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.
Paper Structure (26 sections, 29 theorems, 58 equations)

This paper contains 26 sections, 29 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Some combinatorial results
  3. The noncommutative Hilbert scheme
  4. A semi-orthogonal decomposition of the non-commutative Hilbert scheme
  5. Related work
  6. Acknowledgement
  7. Zonotopes
  8. Generalities
  9. Space tiling zonotopes
  10. The structure of ${\Delta^{m,n}_\tau}$
  11. Proof of Lemma \ref{['lem:admissible']}
  12. Proof of Proposition \ref{['prop:lattice']}
  13. Proof of Proposition \ref{['prop:tilting']}
  14. A semi-orthogonal decomposition for the noncommutative Hilbert scheme
  15. A second proof of the numerical claim in Corollary \ref{['cor:cor1']}
  16. ...and 11 more sections

Key Result

Lemma 1.2

(see §sec:admissible) $\tau$ is admissible if and only if $\partial(\Delta^{m,n}_\tau)\cap {\mathbb Z}^n=\emptyset$.

Theorems & Definitions (54)

  • Definition 1.1
  • Lemma 1.2
  • Proposition 1.3
  • Corollary 1.4
  • Lemma 1.5
  • Corollary 1.6
  • Proposition 1.7: ReinekeMR1048420
  • Proposition 1.8
  • Proposition 1.9
  • Remark 1.10
  • ...and 44 more