Generic Variables in Acyclic Cluster Algebras

By G. Dupont. 2010

67 citations recorded by the source corpus.

Let $Q$ be an acyclic quiver. We introduce the notion of generic variables for the coefficient-free acyclic cluster algebra $\mathcal A(Q)$. We prove that the set $\mathcal G(Q)$ of generic variables contains naturally the set $\mathcal M(Q)$ of cluster monomials in $\mathcal A(Q)$ and that these two sets coincide if and only if $Q$ is a Dynkin quiver. We establish multiplicative properties of these generic variables analogous to multiplicative properties of Lusztig's dual semicanonical basis. This allows to compute explicitly the generic variables when $Q$ is a quiver of affine type. When $Q$ is the Kronecker quiver, the set $\mathcal G(Q)$ is a $\mathbb Z$-basis of $\mathcal A(Q)$ and this basis is compared to Sherman-Zelevinsky and Caldero-Zelevinsky bases.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5b2b7ebf120678b4e4ea366ad44c56568cbfd914

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Citations in the corpus

Quiver varieties and cluster algebras (2009, background): “This, at first sight, looks similar to the set of generic variables considered by Dupont [20].”
Cluster algebras and derived categories (2012, background): “(16) Notice that in total, we find 6 distinct c-vectors and that these are in natural bijection with the (positive and negative) roots of the root system corresponding to the underlying graph A2 of the quiver Q: We simply map a c-vector with components c1 and c2 to the root c1α1 + c2α2, where α1…”
A survey on maximal green sequences (2019, background): “Generic bases were first considered by Dupont [38] for acyclic quivers.”