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Quivers with potentials for Grassmannian cluster algebras

Part of: Arithmetic rings and other special rings Representation theory of rings and algebras

Published online by Cambridge University Press:  21 June 2022

Wen Chang Open the ORCID record for Wen Chang [Opens in a new window]  and
Jie Zhang
Wen Chang
Affiliation:
School of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China e-mail: changwen161@163.com
Jie Zhang*
Affiliation:
School of Mathematics and Statistics, Beijing Institute of Technology, 100081 Beijing, P. R. China
*
e-mail: jiezhang@bit.edu.cn
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Abstract

We consider a quiver with potential (QP) $(Q(D),W(D))$ and an iced quiver with potential (IQP) $(\overline {Q}(D), F(D), \overline {W}(D))$ associated with a Postnikov Diagram D and prove that their mutations are compatible with the geometric exchanges of D. This ensures that we may define a QP $(Q,W)$ and an IQP $(\overline {Q},F,\overline {W})$ for a Grassmannian cluster algebra up to mutation equivalence. It shows that $(Q,W)$ is always rigid (thus nondegenerate) and Jacobi-finite. Moreover, in fact, we show that it is the unique nondegenerate (thus rigid) QP by using a general result of Geiß, Labardini-Fragoso, and Schröer (2016, Advances in Mathematics 290, 364–452).

Then we show that, within the mutation class of the QP for a Grassmannian cluster algebra, the quivers determine the potentials up to right equivalence. As an application, we verify that the auto-equivalence group of the generalized cluster category ${\mathcal {C}}_{(Q, W)}$ is isomorphic to the cluster automorphism group of the associated Grassmannian cluster algebra ${{\mathcal {A}}_Q}$ with trivial coefficients.

Keywords

Quiver with potentialGrassmannian cluster algebracluster category

MSC classification

Primary: 13F60: Cluster algebras
Secondary: 16G20: Representations of quivers and partially ordered sets
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 4 , August 2023 , pp. 1199 - 1225
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

Wen Chang is supported by the NSF of China (Grant No. 11601295), Shaanxi Province, and Shaanxi Normal University. Jie Zhang is supported by the NSF of China (Grant Nos. 12071026 and 12122101), Beijing, and Beijing Institute of Technology.

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