Generic bases for cluster algebras and the Chamber Ansatz

By C. Geiss, Bernard Leclerc, J. Schröer. 2010

107 citations recorded by the source corpus.

Let Q be a finite quiver without oriented cycles, and let $\Lambda$ be the corresponding preprojective algebra. Let g be the Kac-Moody Lie algebra with Cartan datum given by Q, and let W be its Weyl group. With w in W is associated a unipotent cell N^w of the Kac-Moody group with Lie algebra g. In previous work we proved that the coordinate ring \C[N^w] of N^w is a cluster algebra in a natural way. A central role is played by generating functions \vphi_X of Euler characteristics of certain varieties of partial composition series of X, where X runs through all modules in a Frobenius subcategory C_w of the category of nilpotent $\Lambda$-modules. We show that for every X in C_w, \vphi_X coincides after appropriate changes of variables with the cluster characters of Fu and Keller associated with any cluster-tilting module T of C_w. As an application, we get a new description of a generic basis of the cluster algebra obtained from \C[N^w] via specialization of coefficients to 1. For the special case of coefficient-free acyclic cluster algebras this proves a conjecture by Dupont.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/02c9632648159dfb8a9b1c4526314e5e54dc83b7

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Tensor diagrams and cluster algebras (2012, background): “It is then natural to ask: Does the web basis always coincide with the appropriate dual semicanonical (or “generic”) basis, in the sense of [17, 20]?”