Generalized cluster complexes and Coxeter combinatorics

Abstract

We introduce and study a family of simplicial complexes associated to an arbitrary finite root system and a nonnegative integer parameter m. For m=1, our construction specializes to the (simplicial) generalized associahedra or, equivalently, to the cluster complexes for the cluster algebras of finite type. Our computation of the face numbers and h-vectors of these complexes produces the enumerative invariants defined in other contexts by C.A.Athanasiadis, suggesting links to a host of well studied problems in algebraic combinatorics of finite Coxeter groups, root systems, and hyperplane arrangements. Recurrences satisfied by the face numbers of our complexes lead to combinatorial algorithms for determining Coxeter-theoretic invariants. That is, starting with a Coxeter diagram of a finite Coxeter group, one can compute the Coxeter number, the exponents, and other classical invariants by a recursive procedure that only uses most basic graph-theoretic concepts applied to the input diagram. In types A and B, we rediscover the constructions and results obtained by E.Tzanaki <math.CO/0501100>.

The only change in this version is that the introduction was made into a numbered section. This was done to make the arXiv version's section numbering agree with the section numbering in the published version. 40 pages

References in corpus (1)

• Polygon dissections and some generalizations of cluster complexes

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• Generalized frieze pattern determinants and higher angulations of polygons
• The $M$-triangle of generalised non-crossing partitions for the types $E_7$ and $E_8$
• Algèbres amassées et applications
• Counting chains in the noncrossing partition lattice via the W-Laplacian
• Doppelgängers: Bijections of Plane Partitions
• Minuscule doppelgängers, the coincidental down-degree expectations property, and rowmotion
• Positive Fuss-Catalan numbers and Simple-minded systems in negative Calabi-Yau categories
• The EKR property for flag pure simplicial complexes without boundary
• Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions
• Cluster categories, selfinjective algebras, and stable Calabi-Yau dimensions: type A
• Defining an m-cluster category
• Derived equivalence classification of m-cluster tilted algebras of type A
• The geometry of Brauer graph algebras and cluster mutations
• Tamari Lattices for Parabolic Quotients of the Symmetric Group
• A geometric description of $m$-cluster categories
• Repetitive higher cluster categories of type A_n
• Lyashko-Looijenga morphisms and submaximal factorisations of a Coxeter element
• On relations for zeros of $f$-polynomials and $f^{+}$-polynomials
• The Real-rootedness of Generalized Narayana Polynomials
• Faces of generalized cluster complexes and noncrossing partitions
• Coloured quiver mutation for higher cluster categories
• Coloured quivers of type A and the cell-growth problem
• Groupes de réflexion, géométrie du discriminant et partitions non-croisées
• Cluster combinatorics of d-cluster categories
• Combinatorial model for m-cluster categories in type E
• The permuto-associahedron revisited
• Coxeter-biCatalan combinatorics
• Refined enumeration of noncrossing chains and hook formulas
• A circular order on edge-coloured trees and RNA m-diagrams
• Shellability and higher Cohen-Macaulay connectivity of generalized cluster complexes
• A generalization of Euler numbers to finite Coxeter groups
• An introduction to higher cluster categories
• On the enumeration of positive cells in generalized cluster complexes and Catalan hyperplane arrangements
• Hochschild cohomology of $m$-Cluster tilted algebras of type $\tilde{\mathbb{A}}$
• Facets of the $m$-generalized cluster complex and regions in the $m$-extended Catalan arrangement of type $A_n$
• Cyclic sieving and cluster multicomplexes
• The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces
• Fuss-Schröder Paths and Rooted Plane Forests
• A combinatorial reciprocity theorem for hyperplane arrangements
• Dominant regions in noncrystallographic hyperplane arrangements
• Horizontal and vertical mutation fans
• Alexander Duality and Rational Associahedra
• SB-Labelings, Distributivity, and Bruhat Order on Sortable Elements
• On the H-triangle of generalised nonnesting partitions
• The $m$-Cover Posets and Their Applications