Quantum cluster characters for valued quivers

Rupel, Dylan

doi:10.1090/S0002-9947-2015-06251-5

Quantum cluster characters for valued quivers
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by Dylan Rupel PDF

Trans. Amer. Math. Soc. 367 (2015), 7061-7102 Request permission

Abstract:

Let $\mathbb {F}$ be a finite field and $(Q,\mathbf {d})$ an acyclic valued quiver with associated exchange matrix $\tilde {B}$. We follow Hubery’s approach to prove our main conjecture from 2011: the quantum cluster character gives a bijection from the isoclasses of indecomposable rigid valued representations of $Q$ to the set of non-initial quantum cluster variables for the quantum cluster algebra $\mathcal {A}_{|\mathbb {F}|}(\tilde {B},\Lambda )$. As a corollary we find that for any rigid valued representation $V$ of $Q$, all Grassmannians of subrepresentations $Gr_{\mathbf {e}}^V$ have counting polynomials.

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