Mutating seeds: types 𝔸 and 𝔸 ˜.
Annales Mathématiques Blaise Pascal, Tome 19 (2012) no. 1, pp. 29-73.

In the cases 𝔸 and 𝔸 ˜, we describe the seeds obtained by sequences of mutations from an initial seed. In the 𝔸 ˜ 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.

DOI : https://doi.org/10.5802/ambp.304
Classification : 13F60,  16G20,  16G99
Mots clés: Cluster algebras, mutations, seeds, quivers
@article{AMBP_2012__19_1_29_0,
     author = {Ibrahim Assem and Christophe Reutenauer},
     title = {Mutating seeds: types $\mathbb{A}$ and $\widetilde{\mathbb{A}}$.},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {29--73},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {19},
     number = {1},
     year = {2012},
     doi = {10.5802/ambp.304},
     mrnumber = {2978313},
     zbl = {1259.13013},
     language = {en},
     url = {ambp.centre-mersenne.org/item/AMBP_2012__19_1_29_0/}
}
Ibrahim Assem; Christophe Reutenauer. Mutating seeds: types $\mathbb{A}$ and $\widetilde{\mathbb{A}}$.. Annales Mathématiques Blaise Pascal, Tome 19 (2012) no. 1, pp. 29-73. doi : 10.5802/ambp.304. https://ambp.centre-mersenne.org/item/AMBP_2012__19_1_29_0/

[1] Eiichi Abe Hopf algebras, Cambridge Tracts in Mathematics, Volume 74, Cambridge University Press, Cambridge, 1980 (Translated from the Japanese by Hisae Kinoshita and Hiroko Tanaka) | MR 594432 | Zbl 0476.16008

[2] I. Assem; T. Brüstle; R. Schiffler Cluster-tilted algebras as trivial extensions, Bull. Lond. Math. Soc., Volume 40 (2008) no. 1, pp. 151-162 | Article | MR 2409188

[3] Ibrahim Assem; Thomas Brüstle; Gabrielle Charbonneau-Jodoin; Pierre-Guy Plamondon Gentle algebras arising from surface triangulations, Algebra Number Theory, Volume 4 (2010) no. 2, pp. 201-229 | Article | MR 2592019

[5] Ibrahim Assem; Grégoire Dupont; Ralf Schiffler; David Smith Friezes, strings and cluster variables, Glasg. Math. J., Volume 54 (2012) no. 1, pp. 27-60 | Article | MR 2862382

[6] Ibrahim Assem; Christophe Reutenauer; David Smith Friezes, Adv. Math., Volume 225 (2010) no. 6, pp. 3134-3165 | Article | MR 2729004

[7] J. Bastian Mutation classes of A ˜ n -quivers and derived equivalence classification of cluster tilted algebras of type A ˜ n (arXiv:0901.1515v5, to appear)

[8] K. Baur; R. March Categorification of a frieze pattern determinant (arXiv:1008.5329v1)

[9] François Bergeron; Christophe Reutenauer SL k -tilings of the plane, Illinois J. Math., Volume 54 (2010) no. 1, pp. 263-300 http://projecteuclid.org/getRecord?id=euclid.ijm/1299679749 | MR 2776996

[10] Jean Berstel; Christophe Reutenauer Noncommutative rational series with applications, Encyclopedia of Mathematics and its Applications, Volume 137, Cambridge University Press, Cambridge, 2011 | MR 2760561

[11] Aslak Bakke Buan; Robert Marsh; Markus Reineke; Idun Reiten; Gordana Todorov Tilting theory and cluster combinatorics, Adv. Math., Volume 204 (2006) no. 2, pp. 572-618 | Article | MR 2249625

[12] Aslak Bakke Buan; Robert J. Marsh; Idun Reiten Cluster mutation via quiver representations, Comment. Math. Helv., Volume 83 (2008) no. 1, pp. 143-177 | Article | MR 2365411

[13] Aslak Bakke Buan; Dagfinn F. Vatne Derived equivalence classification for cluster-tilted algebras of type A n , J. Algebra, Volume 319 (2008) no. 7, pp. 2723-2738 | Article | MR 2397404

[14] Philippe Caldero; Frédéric Chapoton Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv., Volume 81 (2006) no. 3, pp. 595-616 | Article | MR 2250855

[15] Philippe Caldero; Bernhard Keller From triangulated categories to cluster algebras, Invent. Math., Volume 172 (2008) no. 1, pp. 169-211 | Article | MR 2385670

[16] P. M. Cohn Free rings and their relations, London Mathematical Society Monographs, Volume 19, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1985 | MR 800091 | Zbl 0659.16001

[17] J. H. Conway; H. S. M. Coxeter Triangulated polygons and frieze patterns, Math. Gaz., Volume 57 (1973) no. 400, pp. 87-94 | Article | MR 461269 | Zbl 0285.05028

[18] J. H. Conway; H. S. M. Coxeter Triangulated polygons and frieze patterns, Math. Gaz., Volume 57 (1973) no. 401, pp. 175-183 | Article | MR 461270 | Zbl 0288.05021

[19] H. S. M. Coxeter Frieze patterns, Acta Arith., Volume 18 (1971), pp. 297-310 | MR 286771 | Zbl 0217.18101

[20] Sorin Dăscălescu; Constantin Năstăsescu; Şerban Raianu Hopf algebras, Monographs and Textbooks in Pure and Applied Mathematics, Volume 235, Marcel Dekker Inc., New York, 2001 (An introduction) | MR 1786197

[21] G. Dupont Cluster multiplication in regular components via generalized Chebyshev polynomials (Algebras and Representation Theory, in press)

[22] G. Dupont Generalized Chebyshev Polynomials and Positivity for Regular Cluster Characters (arXiv:0911.0714)

[23] G. Dupont Quantized Chebyshev polynomials and cluster characters with coefficients, J. Algebraic Combin., Volume 31 (2010) no. 4, pp. 501-532 | Article | MR 2639723

[24] Sergey Fomin; Michael Shapiro; Dylan Thurston Cluster algebras and triangulated surfaces. I. Cluster complexes, Acta Math., Volume 201 (2008) no. 1, pp. 83-146 | Article | MR 2448067

[25] Sergey Fomin; Andrei Zelevinsky Cluster algebras. I. Foundations, J. Amer. Math. Soc., Volume 15 (2002) no. 2, p. 497-529 (electronic) | Article | MR 1887642

[26] Sergey Fomin; Andrei Zelevinsky Cluster algebras. II. Finite type classification, Invent. Math., Volume 154 (2003) no. 1, pp. 63-121 | Article | MR 2004457

[27] A. Fordy; R. Marsh Cluster mutation-periodic quivers and associated Laurent sequences (arXiv:0904.0200v3)

[28] Ronald L. Graham; Donald E. Knuth; Oren Patashnik Concrete mathematics, Addison-Wesley Publishing Company, Reading, MA, 1994 (A foundation for computer science) | MR 1397498 | Zbl 0668.00003

[29] Dieter Happel; Claus Michael Ringel Construction of tilted algebras, Representations of algebras (Puebla, 1980) (Lecture Notes in Math.) Volume 903, Springer, Berlin, 1981, pp. 125-144 | MR 654707 | Zbl 0503.16025

[30] Gerhard P. Hochschild Basic theory of algebraic groups and Lie algebras, Graduate Texts in Mathematics, Volume 75, Springer-Verlag, New York, 1981 | MR 620024 | Zbl 0589.20025

[31] Bernhard Keller Cluster algebras, quiver representations and triangulated categories, Triangulated categories (London Math. Soc. Lecture Note Ser.) Volume 375, Cambridge Univ. Press, Cambridge, 2010, pp. 76-160 | MR 2681708

[32] Moss E. Sweedler Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969 | MR 252485 | Zbl 0194.32901

[33] Ernest B. Vinberg Linear representations of groups, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2010 (Translated from the 1985 Russian original by A. Iacob, Reprint of the 1989 translation) | MR 2761806

[34] J.H.M. Wedderburn Non-commutative domains of integrity, J. Reine Angew. Math., Volume 167 (1932), pp. 129-141 | Article | Zbl 0003.20103