Generalized cluster complexes and Coxeter combinatorics
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2005
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| _version_ | 1866913119218958336 |
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| author | Fomin, Sergey Reading, Nathan |
| author_facet | Fomin, Sergey Reading, Nathan |
| contents | 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>. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0505085 |
| institution | arXiv |
| publishDate | 2005 |
| record_format | arxiv |
| spellingShingle | Generalized cluster complexes and Coxeter combinatorics Fomin, Sergey Reading, Nathan Combinatorics Representation Theory 20F55 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>. |
| title | Generalized cluster complexes and Coxeter combinatorics |
| topic | Combinatorics Representation Theory 20F55 |
| url | https://arxiv.org/abs/math/0505085 |