Abstract
We study a finite-dimensional algebra Λ from a Postnikov diagram D in a disk, obtained from the dimer algebra of Baur-King-Marsh by factoring out the ideal generated by the boundary idempotent. Thus, Λ is isomorphic to the stable endomorphism algebra of a cluster tilting module T ∈CM(B) introduced by Jensen-King-Su in order to categorify the cluster algebra structure of \(\mathbb {C}[\text {Gr}_{k}(\mathbb {C}^{n})]\). We show that Λ is self-injective if and only if D has a certain rotational symmetry. In this case, Λ is the Jacobian algebra of a self-injective quiver with potential, which implies that its truncated Jacobian algebras in the sense of Herschend-Iyama are 2-representation finite. We study cuts and mutations of such quivers with potential leading to some new 2-representation finite algebras.
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References
Baur, K., Bogdanic, D.: Extensions between Cohen–Macaulay modules of Grassmanniann cluster categories. J. Algebraic Combin. 45(4), 965–1000 (2017)
Baur, K., Bogdanic, D., Elsener, A.G.: Cluster categories from Grassmannians and root combinatorics. arXiv:1807.05181 (2018)
Buan, A.B., Iyama, O., Reiten, I., Smith, D.: Mutation of cluster-tilting objects and potentials. Amer. J. Math. 133(4), 835–887 (2011)
Baur, K., King, A.D., Marsh, R.J.: Dimer models and cluster categories of Grassmannians. Proc. Lond. Math. Soc. (3) 113(2), 213–260 (2016)
Bocklandt, R: A dimer ABC. Bull. Lond. Math. Soc. 48(3), 387–451 (2016)
Demonet, L., Luo, X.: Ice quivers with potential associated with triangulations and Cohen-Macaulay modules over orders. Trans. Amer. Math. Soc. 368(6), 4257–4293 (2016)
Geiss, C., Leclerc, B., Schröer, J.: Partial flag varieties and preprojective algebras. Ann. Inst. Fourier (Grenoble) 58(3), 825–876 (2008)
Herschend, M., Iyama, O.: n-representation-finite algebras and twisted fractionally Calabi-Yau algebras. Bull. Lond. Math. Soc. 43(3), 449–466 (2011)
Herschend, M., Iyama, O.: Selfinjective quivers with potential and 2-representation-finite algebras. Compos. Math. 147(6), 1885–1920 (2011)
Iyama, O., Oppermann, S.: Stable categories of higher preprojective algebras. Adv Math. 244, 23–68 (2013)
Iyama, O.: Auslander-Reiten theory revisited. In: Trends in Representation Theory of Algebras and Related Topics, EMS Ser. Congr. Rep., pp 349–397. Eur. Math. Soc., Zürich (2008)
Jensen, B.T., King, A.D., Su, X.: A categorification of Grassmannian cluster algebras. Proc. Lond. Math Soc. (3) 113(2), 185–212 (2016)
Marsh, R.J., Rietsch, K.: The B-model connection and mirror symmetry for grassmannians. arXiv:1307.1085 (2013)
Suho, O., Postnikov, A, Speyer, D.E.: Weak separation and plabic graphs. Proc. Lond. Math. Soc. (3) 110(3), 721–754 (2015)
Postnikov, A.: Total positivity, Grassmannians, and networks.arXiv:math/0609764v1 (2006)
Pressland, M.: Mutation of frozen Jacobian algebras. arXiv:1810.01179 (2018)
Ringel, C.M.: The self-injective cluster-tilted algebras. Arch. Math. (Basel) 91 (3), 218–225 (2008)
Scott, J.S.: Grassmannians and cluster algebras. Proc. London Math. Soc. (3) 92(2), 345–380 (2006)
Acknowledgements
I am thankful to my advisor Martin Herschend for the many helpful discussions and comments. I would like to thank Jakob Zimmermann for his help with the computational aspects of determining self-injectivity, and both him and Laertis Vaso for suggestions about the manuscript. I also thank the anonymous referees for spotting issues and suggesting improvements to previous versions of the paper. Finally, my thanks go to Karin Baur, Alastair King and Robert Marsh, for their helpful comments.
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Presented by: Henning Krause
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Pasquali, A. Self-Injective Jacobian Algebras from Postnikov Diagrams. Algebr Represent Theor 23, 1197–1235 (2020). https://doi.org/10.1007/s10468-019-09882-8
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DOI: https://doi.org/10.1007/s10468-019-09882-8