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Quantum cluster characters for valued quivers


Author: Dylan Rupel
Journal: Trans. Amer. Math. Soc. 367 (2015), 7061-7102
MSC (2010): Primary 16G20, 16T99; Secondary 16G70, 16S38
DOI: https://doi.org/10.1090/S0002-9947-2015-06251-5
Published electronically: March 2, 2015
MathSciNet review: 3378824
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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}_{\vert\mathbb{F}\vert}(\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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Additional Information

Dylan Rupel
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Address at time of publication: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Email: drupel@uoregon.edu, d.rupel@neu.edu

DOI: https://doi.org/10.1090/S0002-9947-2015-06251-5
Keywords: Quantum cluster algebra, valued quiver Grassmannian
Received by editor(s): August 29, 2012
Received by editor(s) in revised form: July 24, 2013
Published electronically: March 2, 2015
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.