Combinatorial frameworks for cluster algebras
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arXiv
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| Format: | Preprint |
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2011
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| _version_ | 1866918525058154496 |
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| author | Reading, Nathan Speyer, David E |
| author_facet | Reading, Nathan Speyer, David E |
| contents | We develop a general approach to finding combinatorial models for cluster algebras. The approach is to construct a labeled graph called a framework. When a framework is constructed with certain properties, the result is a model incorporating information about exchange matrices, principal coefficients, g-vectors, and g-vector fans. The idea behind frameworks arises from Cambrian combinatorics and sortable elements, and in this paper, we use sortable elements to construct a framework for any cluster algebra with an acyclic initial exchange matrix. This Cambrian framework yields a model of the entire exchange graph when the cluster algebra is of finite type. Outside of finite type, the Cambrian framework models only part of the exchange graph. In a forthcoming paper, we extend the Cambrian construction to produce a complete framework for a cluster algebra whose associated Cartan matrix is of affine type. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1111_2652 |
| institution | arXiv |
| publishDate | 2011 |
| record_format | arxiv |
| spellingShingle | Combinatorial frameworks for cluster algebras Reading, Nathan Speyer, David E Combinatorics 16S99, 20F55 We develop a general approach to finding combinatorial models for cluster algebras. The approach is to construct a labeled graph called a framework. When a framework is constructed with certain properties, the result is a model incorporating information about exchange matrices, principal coefficients, g-vectors, and g-vector fans. The idea behind frameworks arises from Cambrian combinatorics and sortable elements, and in this paper, we use sortable elements to construct a framework for any cluster algebra with an acyclic initial exchange matrix. This Cambrian framework yields a model of the entire exchange graph when the cluster algebra is of finite type. Outside of finite type, the Cambrian framework models only part of the exchange graph. In a forthcoming paper, we extend the Cambrian construction to produce a complete framework for a cluster algebra whose associated Cartan matrix is of affine type. |
| title | Combinatorial frameworks for cluster algebras |
| topic | Combinatorics 16S99, 20F55 |
| url | https://arxiv.org/abs/1111.2652 |