Acyclic Cluster Algebras Revisited

By David E. Speyer, Hugh Thomas. 2012

50 citations recorded by the source corpus.

We describe a new way to relate an acyclic, skew-symmetrizable cluster algebra to the representation theory of a finite dimensional hereditary algebra. This approach is designed to explain the c-vectors of the cluster algebra. We obtain a necessary and sufficient combinatorial criterion for a collection of vectors to be the c-vectors of some cluster in the cluster algebra associated to a given skew-symmetrizable matrix. Our approach also yields a simple proof of the known result that the c-vectors of an acyclic cluster algebra are sign-coherent, from which Nakanishi and Zelevinsky have showed that it is possible to deduce in an elementary way several important facts about cluster algebras.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e09ac007ef1daafb613a9ad1efd900833c124722

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Citations in the corpus

Silting objects, simple-minded collections, $t$-structures and co-$t$-structures for finite-dimensional algebras (2012, background): “6] and our bijections. The link studied by King and Qiu can be expected to generalise, to some extent, to ﬁnite-dimensional Jacobian algebras. This will help to extend the result of Speyer and Thomas [66]. 1.2. An application. Theorem 7.1 implies the existence of bounded t-structures with unusual pi”
Cluster algebras and derived categories (2012, unknown): “We simply map a c-vector with components c1 and c2 to the root c1α1 + c2α2, where α1 and α2 are the simple roots. ◦ α1 α2 // ☞FF ☞☞ ☞☞ ☞ X✷✷X ✷✷ ✷✷ oo ☞☞ ☞☞ ☞☞ ✷˘˘ ✷✷ ✷✷ ✷ As shown in [122], cf. also [135], this bijection generalizes to all cluster-ﬁnite cluster algebras. In particular, we see that ”
Ordered Exchange Graphs (2013, unknown): “For more details on c-vectors, we refer the reader to [35] where they were introduced and to [84, 83, 98, 82, 66] where they were studied.”