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arXiv · 2509.07883

Oriented matroids from type $\mathbb{A}$ cluster categories

Nicholas J. Williams
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Abstract

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.

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Nicholas J. Williams. 2025-09-09. Oriented matroids from type $\mathbb{A}$ cluster categories. https://arxiv.org/abs/2509.07883

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