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arXiv · 1404.4260

C-vectors via $τ$-tilting theory

Changjian Fu
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Abstract

Inspired by the tropical duality in cluster algebras, we introduce c-vectors for finite-dimensional algebras via $τ$-tilting theory. Let $A$ be a finite-dimensional algebra over a field $k$. Each c-vector of $A$ can be realized as the (negative) dimension vector of certain indecomposable $A$-module and hence we establish the sign-coherence property of this kind of $c$-vectors. We then study the positive c-vectors for certain classes of finite-dimensional algebras. More precisely, we establish the equalities between the set of positive c-vectors and the set of dimension vectors of exceptional modules for quasitilted algebras and representation-directed algebras respectively. This generalizes the equalitites of c-vectors for acyclic cluster algebras obtained by Chávez. To this end, a short proof for the sign-coherence of c-vectors for skew-symmetric cluster algebras has been given in the appendix.

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Changjian Fu. 2015-10-29. C-vectors via $τ$-tilting theory. https://doi.org/10.1016/j.jalgebra.2016.10.031

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