Mutating seeds: types $\mathbb{A}$ and $\widetilde{\mathbb{A}}$.

Assem, Ibrahim; Reutenauer, Christophe

doi:10.5802/ambp.304

Mutating seeds: types

𝔸

and

\tilde{𝔸}

Assem, Ibrahim ¹ ; Reutenauer, Christophe ²

¹ Département de mathématiques, Université de Sherbrooke Sherbrooke (Québec) J1K2R1,Canada
² Laboratoire de combinatoire et d’informatique mathématique, Université du Québec à Montréal Case postale 8888, succursale Centre-ville Montréal (Québec) H3C 3P8, Canada

Annales mathématiques Blaise Pascal, Tome 19 (2012) no. 1, pp. 29-73.

Résumé

In the cases $𝔸$ and $\tilde{𝔸}$ , we describe the seeds obtained by sequences of mutations from an initial seed. In the $\tilde{𝔸}$ 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.

MR Zbl

DOI : 10.5802/ambp.304

Classification : 13F60, 16G20, 16G99
Mots clés : Cluster algebras, mutations, seeds, quivers

Affiliations des auteurs :

Assem, Ibrahim ¹ ; Reutenauer, Christophe ²

@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},
     pages = {29--73},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {19},
     number = {1},
     year = {2012},
     doi = {10.5802/ambp.304},
     zbl = {1259.13013},
     mrnumber = {2978313},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/ambp.304/}
}

TY  - JOUR
AU  - Assem, Ibrahim
AU  - Reutenauer, Christophe
TI  - Mutating seeds: types $\mathbb{A}$ and $\widetilde{\mathbb{A}}$.
JO  - Annales mathématiques Blaise Pascal
PY  - 2012
SP  - 29
EP  - 73
VL  - 19
IS  - 1
PB  - Annales mathématiques Blaise Pascal
UR  - http://www.numdam.org/articles/10.5802/ambp.304/
DO  - 10.5802/ambp.304
LA  - en
ID  - AMBP_2012__19_1_29_0
ER  -

%0 Journal Article
%A Assem, Ibrahim
%A Reutenauer, Christophe
%T Mutating seeds: types $\mathbb{A}$ and $\widetilde{\mathbb{A}}$.
%J Annales mathématiques Blaise Pascal
%D 2012
%P 29-73
%V 19
%N 1
%I Annales mathématiques Blaise Pascal
%U http://www.numdam.org/articles/10.5802/ambp.304/
%R 10.5802/ambp.304
%G en
%F 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) no. 1, pp. 29-73. doi : 10.5802/ambp.304. http://www.numdam.org/articles/10.5802/ambp.304/

Bibliographie
Cité par

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

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

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

[4] Assem, Ibrahim; Dupont, Grégoire Friezes and a construction of the Euclidean cluster variables, J. Pure Appl. Algebra, Volume 215 (2011) no. 10, pp. 2322-2340 | DOI | MR

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[31] Keller, Bernhard 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

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

[33] Vinberg, Ernest B. 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

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

Cité par Sources :