In higher Auslander algebras of type A, the d-almost positive subcategory is the d-exangulated quotient of the d-exact subcategory of the module category and the (d+2)-angulated cluster category by ideals from injective-to-projective morphisms.
Oriented matroids from type $\mathbb{A}$ cluster categories
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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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math.RT 1years
2026 1verdicts
UNVERDICTED 1representative citing papers
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Relations between categorifications of higher-dimensional type $A$ cluster combinatorics
In higher Auslander algebras of type A, the d-almost positive subcategory is the d-exangulated quotient of the d-exact subcategory of the module category and the (d+2)-angulated cluster category by ideals from injective-to-projective morphisms.