Indecomposable Modules in the Grassmannian Cluster Category ${\rm CM}(B_{5,10})$
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Authors: D. BOGDANIć AND IVAN-VANJA BOROJA
DOI: 10.46793/KgJMat2406.907B
Abstract:
In this paper, we study indecomposable rank 2 modules in the Grassmannian cluster category CM(B5,10). This is the smallest wild case containing modules whose profile layers are 5-interlacing. We construct all rank 2 indecomposable modules with a specific natural filtration, classify them up to isomorphism, and parameterize all infinite families of non-isomorphic rank 2 modules.
Keywords:
Cohen-Macaulay modules, Grassmannian cluster categories, Indecomposable modules.
References:
[1] I. Assem, D. Simson and A. Skowronski, Elements of the Representation Theory of Associative Algebras: Techniques of Representation Theory, London Mathematical Society Student Texts 1, Cambridge University Press, Cambridge, 2006.
[2] K. Baur and D. Bogdanic, Extensions between Cohen-Macaulay modules of Grassmannian cluster categories, J. Algebraic Combin. 4 (2016), 1–36. https://doi.org/10.1007/s10801-016-0731-5
[3] K. Baur, D. Bogdanic and A. G. Elsener, Cluster categories from Grassmannians and root combinatorics, Nagoya Math. J. 240 (2020), 322–354. https://doi.org/10.1017/nmj.2019.14
[4] K. Baur, D. Bogdanic and J.-R. Li, Construction of rank 2 indecomposable modules in Grassmannian cluster categories, in: The McKay Correspondence, Mutation and Related Topics, Advanced Studies in Pure Mathematics 88, Mathematical Society of Japan, Tokyo, 2022. (to appear)
[5] K. Baur, A. D. King and B. R. Marsh, Dimer models and cluster categories of Grassmannians, Proc. Lond. Math. Soc. (3) 113(2) (2016), 185–212. https://doi.org/10.1002/pamm.201610436
[6] H. Derksen, J. Weyman and A. Zelevinsky, Quivers with potentials and their representations. I. Mutations, Selecta Math. (N.S.) 14(1) (2008), 59–119. https://doi.org/10.1007/s00029-008-0057-9
[7] S. Fomin and A. Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154(1) (2003), 63–121. https://doi.org/10.1007/s00222-003-0302-y
[8] C. Fraser, Braid group symmetries of Grassmannian cluster algebras, Selecta Math. (N.S.) 26 (2020), Paper ID 17, 51 pages. https://doi.org/10.1007/s00029-020-0542-3
[9] C. Geiss, B. Leclerc and J. Schröer, Rigid modules over preprojective algebras, Invent. Math. 165(3) (2006), 589–632. https://doi.org/10.1007/s00222-006-0507-y
[10] C. Geiss, B. Leclerc and J. Schröer, Preprojective algebras and cluster algebras, in: Trends in Representation Theory of Algebras and Related Topics, European Mathematical Society, Zurich, 2008, 253–283. https://doi.org/10.4171/062-1/6
[11] C. Geiss, B. Leclerc and J. Schröer, Partial flag varieties and preprojective algebras, Ann. Inst. Fourier (Grenoble) 58(3) (2008), 825–876. https://doi.org/10.5802/aif.2371
[12] M. Gekhtman, M. Shapiro, A. Stolin and A. Vainshtein, Poisson structures compatible with the cluster algebra structure in Grassmannians, Lett. Math. Phys. 100(2) (2012), 139–150. https://doi.org/10.1007/s11005-012-0547-8
[13] D. Hernandez and B. Leclerc, Cluster algebras and quantum affine algebras, Duke Math. J. 154(2) (2010), 265–341. https://doi.org/10.1215/00127094-2010-040
[14] B. T. Jensen, A. D. King and X. Su, A categorification of Grassmannian cluster algebras, Proc. Lond. Math. Soc. (3) 113(2) (2016), 185–212. https://doi.org/10.1112/plms/pdw028
[15] B. R. Marsh and K. Rietsch, The B-model connection and mirror symmetry for Grassmannians, Adv. Math. 366 (2020), Paper ID 107027, 131 pages. https://doi.org/10.1016/j.aim.2020.107027
[16] G. Muller and D. Speyer, Cluster algebras of Grassmannians are locally acyclic, Proc. Amer. Math. Soc. 144(8) (2016), 3267–3281. https://doi.org/10.1090/proc/13023
[17] J. Scott, Grassmannians and cluster algebras, Proc. Lond. Math. Soc. (3) 92(2) (2006), 345–380. https://doi.org/10.1112/S0024611505015571
[18] D. Speyer and L. Williams, The tropical totally positive Grassmannian, J. Algebraic Combin. 22(2) (2005), 189–210. https://doi.org/10.1007/s10801-005-2513-3
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