Oriented matroids and type $\mathbb{A}$ cluster categories
Nicholas J. Williams
TL;DR
This work connects cluster-algebra mutations to the combinatorics of oriented matroids by constructing, for each basic cluster-tilting object $\mathsf{T}$ in the type $\mathbb{A}_{n}$ cluster category $\mathscr{C}_{n}$, a rank-$4$ oriented matroid $\mathcal{M}_{\mathsf{T}}$ whose stackable triangulations encode equivalence classes of maximal green sequences with initial cluster $\mathsf{T}$. The construction hinges on an extriangulated structure that makes $\mathsf{T}$ projective and yields a well-defined chirotope $\chi_{T}$ from a triangulation $T$ of the $(n+3)$-gon; the main results prove that $\mathcal{M}_{\mathsf{T}}$ is a valid oriented matroid and that there is a bijection between stackable triangulations of $\mathcal{M}_{\mathsf{T}}$ and MG-sequence classes. The paper also provides explicit directions between MG sequences and triangulations and discusses realizability conjectures, generalizing prior work linking MG sequences of linearly oriented $\mathbb{A}_{n}$ to triangulations of 3D cyclic polytopes. The framework offers a polytopal and matroidal lens on MG sequences in type $\mathbb{A}$, with potential realizability by a polytope and broad implications for the combinatorics of cluster mutations.
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.