Generalized cluster complexes and Coxeter combinatorics
By Sergey Fomin, Nathan Reading. 2005
159 citations recorded by the source corpus.
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 .
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/f8e2ce087e617b87e7f339a281d83ad253adc9ee
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Citations in the corpus
- A geometric description of m-cluster categories (2006, background): “SH 4.1. m-coloured almost positive roots and m-diagonals. For Φ a root system, with positive roots Φ+ and simple roots α1,α2,...,αn, let Φm ≥−1 denote the set of m-coloured almost positive roots (see [FR]). An element of Φm ≥−1 is either a m-coloured positive root αk where α∈ Φ+ and k∈ {1,2,...,m} o”
- Triangulated Categories: Cluster algebras, quiver representations and triangulated categories (2008, unknown): “[70] [71] ..., •integrable systems [55] ..., •higher Teichmu¨ller spaces [47] [44] [45] [46] ..., •combinatorics and the study of combinatorial polyhedra like the Stasheff associahedra [34] [33] [100] [49] [109] [50] ..., •commutative and non commutative algebraic geometry, in particular the study of”
- Coloured quiver mutation for higher cluster categories (2008, methodology): “Every codimension 1 face of �m is contained in exactly m + 1 maximal faces [ FR , Proposition 2.10].”
- Surveys in Combinatorics 2011: The cyclic sieving phenomenon: a survey (2010, background): “In fact, Eu and Fu strengthened this theorem in two ways: by looking at the faces of dimension k and by considering an m-divisible generalization of the cluster complex due to Fomin and Reading [20].”
- Cluster-tilted algebras are Gorenstein and stably Calabi–Yau (2005, unknown): “ngles. One shows that the kernel of F|U is (ST ) as in section 2.CLUSTER TILTED ALGEBRAS 15 5.2. Examples of d-cluster tilting subcategories. One can use proposition 5.6 below (cf. also [48] and [47] [20]) to produce the following examples: Consider the algebra Agiven by the quiver 1 //2 3 OO oo 4 ”
- Defining an m-cluster category (2006, methodology): “t preprint of Iyama and Yoshino [IY]. Our approach is different and more elementary. Clusters We begin with a quick introduction to the combinatorics of clusters. Our presentation is based on [FZ] and [FR]. Let Φ be a crystallographic root system of rank n. (In fact, the assumption that Φ is crystall”
- Noncrossing partitions and the shard intersection order (2009, unknown): “(Cf. [15, Propo-sition 8.3].)”
- Cluster algebras and derived categories (2012, background): “…spaces [35] [36] [37] [38] [39] . . . ; • combinatorics and in particular the study of polyhedra like the Stasheff associahedra [21] [22] [42] [43] [71] [90] [102] [104] [103] [130] . . . ; • commutative and non commutative algebraic geometry and in particular the study of stability…”
- Calabi–Yau triangulated categories (2008, background): “The combinatorics of d-cluster categories with finitely many indecomposables are closely related to those of the generalized Coxeter complexes introduced in [ 20 ].”
- Generalized cluster complexes via quiver representations (2006, methodology): “If (Γ,Ω0) is a valued Dynkin graph with an alternating orientation, then the automorphism R of Φ≥−1 defined by Fomin-Zelevinsky [FZ2] is RΩ0 , hence Rd,Ω0 is the usual one (Rd) defined by Fomin-Reading [FR2].”
- Subword complexes, cluster complexes, and generalized multi-associahedra (2011, unknown): “Reading [18].”