Categorification of sign-skew-symmetric cluster algebras and some conjectures on g-vectors

Abstract

Using the unfolding method given in \cite{HL}, we prove the conjectures on sign-coherence and a recurrence formula respectively of $\mathbf{g}$-vectors for acyclic sign-skew-symmetric cluster algebras. As a following consequence, the conjecture is affirmed in the same case which states that the $\mathbf{g}$-vectors of any cluster form a basis of ${\mathbb{Z}}^n$. Also, the additive categorification of an acyclic sign-skew-symmetric cluster algebra $\mathcal{A}(\Sigma )$ is given, which is realized as $({\mathcal{C}}^{\tilde{Q}},\Gamma )$ for a Frobenius $2$-Calabi-Yau category ${\mathcal{C}}^{\tilde{Q}}$ constructed from an unfolding $(Q,\Gamma )$ of the acyclic exchange matrix $B$ of $\mathcal{A}(\Sigma )$.

12 pages

References in corpus (1)

• Unfolding of acyclic sign-skew-symmetric cluster algebras and applications to positivity and $F$-polynomials

Cited by in corpus (3)

• Study on cluster algebras via abstract pattern and two conjectures on d-vectors and g-vector
• Positivity for quantum cluster algebras from unpunctured orbifolds
• Cluster algebras arising from cluster tubes II: the Caldero-Chapoton map