In the cases and , we describe the seeds obtained by sequences of mutations from an initial seed. In the case, we deduce a linear representation of the group of mutations which contains as matrix entries all cluster variables obtained after an arbitrary sequence of mutations (this sequence is an element of the group). Nontransjective variables correspond to certain subgroups of finite index. A noncommutative rational series is constructed, which contains all this information.
@article{AMBP_2012__19_1_29_0,
author = {Assem, Ibrahim and Reutenauer, Christophe},
title = {Mutating seeds: types $\mathbb{A}$ and $\widetilde{\mathbb{A}}$.},
journal = {Annales math\'ematiques Blaise Pascal},
volume = {19},
year = {2012},
pages = {29-73},
doi = {10.5802/ambp.304},
zbl = {1259.13013},
mrnumber = {2978313},
language = {en},
url = {http://dml.mathdoc.fr/item/AMBP_2012__19_1_29_0}
}
Assem, Ibrahim; Reutenauer, Christophe. Mutating seeds: types $\mathbb{A}$ and $\widetilde{\mathbb{A}}$.. Annales mathématiques Blaise Pascal, Tome 19 (2012) pp. 29-73. doi : 10.5802/ambp.304. http://gdmltest.u-ga.fr/item/AMBP_2012__19_1_29_0/
[1] Hopf algebras, Cambridge University Press, Cambridge, Cambridge Tracts in Mathematics, Tome 74 (1980) (Translated from the Japanese by Hisae Kinoshita and Hiroko Tanaka) | MR 594432 | Zbl 0476.16008
[2] Cluster-tilted algebras as trivial extensions, Bull. Lond. Math. Soc., Tome 40 (2008) no. 1, pp. 151-162 | Article | MR 2409188
[3] Gentle algebras arising from surface triangulations, Algebra Number Theory, Tome 4 (2010) no. 2, pp. 201-229 | Article | MR 2592019
[4] Friezes and a construction of the Euclidean cluster variables, J. Pure Appl. Algebra, Tome 215 (2011) no. 10, pp. 2322-2340 | Article | MR 2793939
[5] Friezes, strings and cluster variables, Glasg. Math. J., Tome 54 (2012) no. 1, pp. 27-60 | Article | MR 2862382
[6] Friezes, Adv. Math., Tome 225 (2010) no. 6, pp. 3134-3165 | Article | MR 2729004
[7] Mutation classes of -quivers and derived equivalence classification of cluster tilted algebras of type (arXiv:0901.1515v5, to appear)
[8] Categorification of a frieze pattern determinant (arXiv:1008.5329v1)
[9] -tilings of the plane, Illinois J. Math., Tome 54 (2010) no. 1, pp. 263-300 http://projecteuclid.org/getRecord?id=euclid.ijm/1299679749 | MR 2776996
[10] Noncommutative rational series with applications, Cambridge University Press, Cambridge, Encyclopedia of Mathematics and its Applications, Tome 137 (2011) | MR 2760561
[11] Tilting theory and cluster combinatorics, Adv. Math., Tome 204 (2006) no. 2, pp. 572-618 | Article | MR 2249625
[12] Cluster mutation via quiver representations, Comment. Math. Helv., Tome 83 (2008) no. 1, pp. 143-177 | Article | MR 2365411
[13] Derived equivalence classification for cluster-tilted algebras of type , J. Algebra, Tome 319 (2008) no. 7, pp. 2723-2738 | Article | MR 2397404
[14] Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv., Tome 81 (2006) no. 3, pp. 595-616 | Article | MR 2250855
[15] From triangulated categories to cluster algebras, Invent. Math., Tome 172 (2008) no. 1, pp. 169-211 | Article | MR 2385670
[16] Free rings and their relations, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, London Mathematical Society Monographs, Tome 19 (1985) | MR 800091 | Zbl 0659.16001
[17] Triangulated polygons and frieze patterns, Math. Gaz., Tome 57 (1973) no. 400, pp. 87-94 | Article | MR 461269 | Zbl 0285.05028
[18] Triangulated polygons and frieze patterns, Math. Gaz., Tome 57 (1973) no. 401, pp. 175-183 | Article | MR 461270 | Zbl 0288.05021
[19] Frieze patterns, Acta Arith., Tome 18 (1971), pp. 297-310 | MR 286771 | Zbl 0217.18101
[20] Hopf algebras, Marcel Dekker Inc., New York, Monographs and Textbooks in Pure and Applied Mathematics, Tome 235 (2001) (An introduction) | MR 1786197
[21] Cluster multiplication in regular components via generalized Chebyshev polynomials (Algebras and Representation Theory, in press)
[22] Generalized Chebyshev Polynomials and Positivity for Regular Cluster Characters (arXiv:0911.0714)
[23] Quantized Chebyshev polynomials and cluster characters with coefficients, J. Algebraic Combin., Tome 31 (2010) no. 4, pp. 501-532 | Article | MR 2639723
[24] Cluster algebras and triangulated surfaces. I. Cluster complexes, Acta Math., Tome 201 (2008) no. 1, pp. 83-146 | Article | MR 2448067
[25] Cluster algebras. I. Foundations, J. Amer. Math. Soc., Tome 15 (2002) no. 2, p. 497-529 (electronic) | Article | MR 1887642
[26] Cluster algebras. II. Finite type classification, Invent. Math., Tome 154 (2003) no. 1, pp. 63-121 | Article | MR 2004457
[27] Cluster mutation-periodic quivers and associated Laurent sequences (arXiv:0904.0200v3)
[28] Concrete mathematics, Addison-Wesley Publishing Company, Reading, MA (1994) (A foundation for computer science) | MR 1397498 | Zbl 0668.00003
[29] Construction of tilted algebras, Representations of algebras (Puebla, 1980), Springer, Berlin (Lecture Notes in Math.) Tome 903 (1981), pp. 125-144 | MR 654707 | Zbl 0503.16025
[30] Basic theory of algebraic groups and Lie algebras, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 75 (1981) | MR 620024 | Zbl 0589.20025
[31] Cluster algebras, quiver representations and triangulated categories, Triangulated categories, Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 375 (2010), pp. 76-160 | MR 2681708
[32] Hopf algebras, W. A. Benjamin, Inc., New York, Mathematics Lecture Note Series (1969) | MR 252485 | Zbl 0194.32901
[33] Linear representations of groups, Birkhäuser/Springer, New York, Modern Birkhäuser Classics (2010) (Translated from the 1985 Russian original by A. Iacob, Reprint of the 1989 translation) | MR 2761806
[34] Non-commutative domains of integrity, J. Reine Angew. Math., Tome 167 (1932), pp. 129-141 | Article | Zbl 0003.20103