×

On the wildness of Cambrian lattices. (English) Zbl 1436.16014

Summary: In this note, we investigate the representation type of the cambrian lattices and some other related lattices. The result is expressed as a very simple trichotomy. When the rank of the underlined Coxeter group is at most 2, the lattices are of finite representation type. When the Coxeter group is a reducible group of type \(\mathbb{A}_{1}^{3}\), the lattices are of tame representation type. In all the other cases they are of wild representation type.

MSC:

16G20 Representations of quivers and partially ordered sets
13F60 Cluster algebras
16G60 Representation type (finite, tame, wild, etc.) of associative algebras

Software:

MathOverflow
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Armstrong, D.: Generalized noncrossing partitions and combinatorics of Coxeter groups. Mem. Amer. Math Soc. 202(949), x + 159 (2009) · Zbl 1191.05095
[2] Baryshnikov, Y.: New developments in singularity theory (Cambridge, 2000), volume 21 of NATO Sci. Ser. II Math. Phys. Chem., pp 65-86. Kluwer Acad. Publ., Dordrecht (2001)
[3] Borceux, F.: Handbook of categorical algebra. 1, volume 50 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1994) · Zbl 0911.18001
[4] Chaptal, N: Objets indécomposables dans certaines catégories de foncteurs. C. R. Acad. Sci. Paris Sér. A-B 268, A934—A936 (1969) · Zbl 0175.29003
[5] Chapoton, F.: Stokes posets and serpent nests. DMTCS 18(3), 1-16 (2016). arXiv:1505.05990 · Zbl 1401.06002
[6] Drozd, Y.: Tame and wild matrix problems. In: Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), volume 832 of Lecture Notes in Math., pp. 242-258. Springer, Berlin-New York (1980) · Zbl 0454.16014
[7] Garver, A., McConville, T.: Oriented flip graphs and noncrossing tree partitions. ArXiv e-prints (2016) · Zbl 1427.05235
[8] Happel, D., Zacharia, D.: Homological properties of piecewise hereditary algebras. J. Algebra 323(4), 1139-1154 (2010) · Zbl 1233.16010 · doi:10.1016/j.jalgebra.2009.11.020
[9] Kleiner, M.: Partially ordered sets of finite type. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 28, 32-41 (1972) · Zbl 0345.06001
[10] Ladkani, S.: Universal derived equivalences of posets. ArXiv e-prints (2007) · Zbl 1127.18005
[11] Ladkani, S: Universal derived equivalences of posets of cluster tilting objects. ArXiv e-prints (2007)
[12] Ladkani, S.: Which canonical algebras are derived equivalent to incidence algebras of posets? . Comm. Algebra 36(12), 4599-4606 (2007) · Zbl 1196.16007 · doi:10.1080/00927870802185989
[13] Lenzing, H., Meltzer, H.: Sheaves on a weighted projective line of genus one and representations of a tubular algebra. In: Proceedings of the Sixth International Conference on Representations of Algebras (Ottawa, ON, 1992), volume 14 of Carleton-Ottawa Math. Lecture Note Ser., p. 25. Carleton Univ., Ottawa (1992) · Zbl 0797.14007
[14] Lenzing, H.: Coxeter transformations associated with finite-dimensional algebras. In: Computational methods for representations of groups and algebras (Essen, 1997), volume 173 of Progr. Math. Birkhäuser, Basel (1999) · Zbl 0941.16007
[15] Leszczyński, Z.: The completely separating incidence algebras of tame representation type. Colloq. Math. 94(2), 243-262 (2002) · Zbl 1057.16009 · doi:10.4064/cm94-2-7
[16] Leszczyński, Z.: Representation-tame incidence algebras of finite posets. Colloq. Math. 96(2), 293-305 (2003) · Zbl 1083.16009 · doi:10.4064/cm96-2-11
[17] Loupias, M.: Indecomposable representations of finite ordered sets. In: Representations of algebras (Proc. Internat. Conf., Carleton Univ., Ottawa, Ont., 1974), pp. 201-209. Lecture Notes in Math., Vol. 488. Springer, Berlin (1975)
[18] Loupias, M.: Représentations indécomposables des ensembles ordonnés finis. In: Séminaire d’Algèbre Non Commutative (année 1974/75), Exp. No. 7, pp. 15 pp. Publ. Math. Orsay, No. 154-7543. Univ. Paris XI, Orsay (1975) · Zbl 0362.16015
[19] Nazarova, L. A.: Partially ordered sets of infinite type. Izv. Akad. Nauk SSSR Ser. Mat. 39(5), 963-991 (1975)
[20] Reading, N: Cambrian lattices. Adv. Math. 205(2), 313-353 (2006) · Zbl 1106.20033 · doi:10.1016/j.aim.2005.07.010
[21] Reading, N.: Sortable elements and Cambrian lattices. Algebra Universalis 56 (3-4), 411-437 (2007) · Zbl 1184.20038 · doi:10.1007/s00012-007-2009-1
[22] Rickard, J.: Why is the A6 preprojective algebra of wild representation type? mathoverflow.net, Question 202259 (2015)
[23] Simson, D.: Linear representations of partially ordered sets and vector space categories, volume 4 of Algebra, Logic and Applications. Gordon and Breach Science Publishers, Montreux (1992) · Zbl 0818.16009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.