Oriented matroids from type $\mathbb{A}$ cluster categories
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912964032856064 |
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| author | Williams, Nicholas J. |
| author_facet | Williams, Nicholas J. |
| contents | For any cluster-tilting object $\mathsf{T}$ in the cluster category $\mathscr{C}_{n}$ of type $\mathbb{A}_{n}$, we construct a rank-four oriented matroid $\mathcal{M}_{\mathsf{T}}$ such that stackable triangulations of $\mathcal{M}_{\mathsf{T}}$ are in bijection with equivalence classes of maximal green sequences with initial cluster $\mathsf{T}$. This generalises the result that equivalence classes of maximal green sequences of linearly oriented $\mathbb{A}_{n}$ are in bijection with triangulations of a three-dimensional cyclic polytope. The definition of the oriented matroid $\mathcal{M}_{\mathsf{T}}$ arises from the extriangulated structure on $\mathscr{C}_{n}$ which makes $\mathsf{T}$ projective. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07883 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Oriented matroids from type $\mathbb{A}$ cluster categories Williams, Nicholas J. Combinatorics Representation Theory Primary: 16G20, Secondary: 52C40, 13F60 For any cluster-tilting object $\mathsf{T}$ in the cluster category $\mathscr{C}_{n}$ of type $\mathbb{A}_{n}$, we construct a rank-four oriented matroid $\mathcal{M}_{\mathsf{T}}$ such that stackable triangulations of $\mathcal{M}_{\mathsf{T}}$ are in bijection with equivalence classes of maximal green sequences with initial cluster $\mathsf{T}$. This generalises the result that equivalence classes of maximal green sequences of linearly oriented $\mathbb{A}_{n}$ are in bijection with triangulations of a three-dimensional cyclic polytope. The definition of the oriented matroid $\mathcal{M}_{\mathsf{T}}$ arises from the extriangulated structure on $\mathscr{C}_{n}$ which makes $\mathsf{T}$ projective. |
| title | Oriented matroids from type $\mathbb{A}$ cluster categories |
| topic | Combinatorics Representation Theory Primary: 16G20, Secondary: 52C40, 13F60 |
| url | https://arxiv.org/abs/2509.07883 |