Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 069, 40 pages      arXiv:2209.12540      https://doi.org/10.3842/SIGMA.2023.069

The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces

Theo Douvropoulos a and Matthieu Josuat-Vergès b
a) University of Massachusetts at Amherst, USA
b) IRIF, CNRS, Université Paris-Cité, France

Received September 27, 2022, in final form September 12, 2023; Published online September 26, 2023

Abstract
The generalized cluster complex was introduced by Fomin and Reading, as a natural extension of the Fomin-Zelevinsky cluster complex coming from finite type cluster algebras. In this work, to each face of this complex we associate a parabolic conjugacy class of the underlying finite Coxeter group. We show that the refined enumeration of faces (respectively, positive faces) according to this data gives an explicit formula in terms of the corresponding characteristic polynomial (equivalently, in terms of Orlik-Solomon exponents). This characteristic polynomial originally comes from the theory of hyperplane arrangements, but it is conveniently defined via the parabolic Burnside ring. This makes a connection with the theory of parking spaces: our results eventually rely on some enumeration of chains of noncrossing partitions that were obtained in this context. The precise relations between the formulas counting faces and the one counting chains of noncrossing partitions are combinatorial reciprocities, generalizing the one between Narayana and Kirkman numbers.

Key words: cluster complex; parking functions; noncrossing partitions; Fuß-Catalan numbers; finite Coxeter groups.

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