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Quantum cluster characters for valued quivers. (English) Zbl 1371.16014
Let $$F$$ be a finite field, and let $$(Q,d)$$ be a valued quiver without oriented cycles. In a previous paper [Int. Math. Res. Not. 2011, No. 14, 3207–3236 (2011; Zbl 1237.16013)], the author gave a quantum analogue of the cluster character for valued quiver representations [P. Caldero and F. Chapoton, Comment. Math. Helv. 81, No. 3, 595–616 (2006; Zbl 1119.16013)], replacing the Euler-Poincaré characteristics by the cardinalities of valued quiver Grassmannians. It was conjectured that the quantum cluster character provides a bijection between exceptional representations $$V$$ of $$(Q,d)$$ and non-initial quantum cluster variables of the corresponding quantum cluster algebra. Using a recent approach of A. Hubery [“Acyclic cluster algebras via Ringel-Hall algebras”, Preprint, http://www.maths.leeds.ac.uk/$$\sim$$ahubery/Cluster.pdf], the conjecture is proved. As a corollary, it follows that counting polynomials exist for the Grassmannians $$\mathrm{Gr}^V_e$$ of subrepresentations in $$V$$ of any type $$e$$.

MSC:
 16G20 Representations of quivers and partially ordered sets
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References:
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