A Cambrian framework for the oriented cycle
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2015
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| _version_ | 1866914609163665408 |
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| author | Reading, Nathan Speyer, David E. |
| author_facet | Reading, Nathan Speyer, David E. |
| contents | This paper completes the project of constructing combinatorial models (called frameworks) for the exchange graph and g-vector fan associated to any exchange matrix B whose Cartan companion is of finite or affine type, using the combinatorics and geometry of Coxeter-sortable elements and Cambrian lattices/fans. Specifically, we construct a framework in the unique non-acyclic affine case, the cyclically oriented n-cycle. In the acyclic affine case, a framework was constructed by combining a copy of the Cambrian fan for B with an antipodal copy of the Cambrian fan for -B. In this paper, we extend this "doubled Cambrian fan" construction to the oriented n-cycle, using a more general notion of sortable elements for quivers with cycles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1506_04210 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | A Cambrian framework for the oriented cycle Reading, Nathan Speyer, David E. Combinatorics This paper completes the project of constructing combinatorial models (called frameworks) for the exchange graph and g-vector fan associated to any exchange matrix B whose Cartan companion is of finite or affine type, using the combinatorics and geometry of Coxeter-sortable elements and Cambrian lattices/fans. Specifically, we construct a framework in the unique non-acyclic affine case, the cyclically oriented n-cycle. In the acyclic affine case, a framework was constructed by combining a copy of the Cambrian fan for B with an antipodal copy of the Cambrian fan for -B. In this paper, we extend this "doubled Cambrian fan" construction to the oriented n-cycle, using a more general notion of sortable elements for quivers with cycles. |
| title | A Cambrian framework for the oriented cycle |
| topic | Combinatorics |
| url | https://arxiv.org/abs/1506.04210 |