×

Mutation classes of skew-symmetrizable \(3\times 3\) matrices. (English) Zbl 1260.05168

Summary: Mutation of skew-symmetrizable matrices is a fundamental operation that first arose in Fomin-Zelevinsky’s theory of cluster algebras; it also appears naturally in many different areas of mathematics.
In this paper, we study mutation classes of skew-symmetrizable \(3\times 3\) matrices and associated graphs. We determine representatives for these classes using a natural minimality condition, generalizing and strengthening results of Beineke-Brustle-Hille and Felikson-Shapiro-Tumarkin. Furthermore, we obtain a new numerical invariant for the mutation operation on skew-symmetrizable matrices of arbitrary size.

MSC:

05E15 Combinatorial aspects of groups and algebras (MSC2010)
05E05 Symmetric functions and generalizations
15B36 Matrices of integers
05C22 Signed and weighted graphs
13F60 Cluster algebras

Software:

quivermutation
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ibrahim Assem, Martin Blais, Thomas Brüstle, and Audrey Samson, Mutation classes of skew-symmetric 3\times 3-matrices, Comm. Algebra 36 (2008), no. 4, 1209 – 1220. · Zbl 1148.16025 · doi:10.1080/00927870701861243
[2] Michael Barot, Christof Geiss, and Andrei Zelevinsky, Cluster algebras of finite type and positive symmetrizable matrices, J. London Math. Soc. (2) 73 (2006), no. 3, 545 – 564. · Zbl 1093.05070 · doi:10.1112/S0024610706022769
[3] Andre Beineke, Thomas Brüstle, and Lutz Hille, Cluster-cylic quivers with three vertices and the Markov equation, Algebr. Represent. Theory 14 (2011), no. 1, 97 – 112. With an appendix by Otto Kerner. · Zbl 1239.16017 · doi:10.1007/s10468-009-9179-9
[4] Philippe Caldero and Bernhard Keller, From triangulated categories to cluster algebras. II, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 6, 983 – 1009 (English, with English and French summaries). · Zbl 1115.18301 · doi:10.1016/j.ansens.2006.09.003
[5] A. Felikson, M. Shapiro and P. Tumarkin, Skew-symmetric cluster algebras of finite mutation type, J. European Math. Soc. 14 (4) (2012), 1135-1180. · Zbl 1262.13038
[6] A. Felikson, M. Shapiro and P. Tumarkin, Cluster algebras of finite mutation type via unfoldings, Int. Math. Res. Notices 2012 (8), 1768-1804. · Zbl 1283.13020
[7] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), no. 1, 63 – 121. · Zbl 1054.17024 · doi:10.1007/s00222-003-0302-y
[8] S. Fomin and A. Zelevinsky, Cluster Algebras: Notes for the CDM-03 conference, Current developments in mathematics, 1-34, Int. Press, Somerville, MA, 2003. · Zbl 1119.05108
[9] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. · Zbl 0716.17022
[10] Bernhard Keller, Cluster algebras, quiver representations and triangulated categories, Triangulated categories, London Math. Soc. Lecture Note Ser., vol. 375, Cambridge Univ. Press, Cambridge, 2010, pp. 76 – 160. · Zbl 1215.16012
[11] B. Keller, Quiver mutation in Java, Java applet available at the author’s home page.
[12] Ahmet I. Seven, Cluster algebras and semipositive symmetrizable matrices, Trans. Amer. Math. Soc. 363 (2011), no. 5, 2733 – 2762. · Zbl 1291.05225
[13] Ahmet I. Seven, Mutation classes of 3\times 3 generalized Cartan matrices, Highlights in Lie algebraic methods, Progr. Math., vol. 295, Birkhäuser/Springer, New York, 2012, pp. 205 – 211. · Zbl 1248.05212 · doi:10.1007/978-0-8176-8274-3_9
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.