Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T05:04:11.618Z Has data issue: false hasContentIssue false
  • English
  • Français

FRIEZES, STRINGS AND CLUSTER VARIABLES

Published online by Cambridge University Press:  02 August 2011

IBRAHIM ASSEM ,
GRÉGOIRE DUPONT ,
RALF SCHIFFLER  and
DAVID SMITH
IBRAHIM ASSEM
Affiliation:
Université de Sherbrooke, Sherbrooke QC, Canada E-mail: ibrahim.assem@usherbrooke.ca
GRÉGOIRE DUPONT
Affiliation:
Université de Sherbrooke, Sherbrooke QC, Canada E-mail: gregoire.dupont@usherbrooke.ca
RALF SCHIFFLER
Affiliation:
University of Connecticut, Storrs CT, USA E-mail: schiffler@math.uconn.edu
DAVID SMITH
Affiliation:
University Bishop's, Sherbrooke QC, Canada E-mail: dsmith@ubishops.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

To any walk in a quiver, we associate a Laurent polynomial. When the walk is the string of a string module over a 2-Calabi–Yau tilted algebra, we prove that this Laurent polynomial coincides with the corresponding cluster character of the string module up to an explicit normalising monomial factor.

Keywords

13F6016G20
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 1 , January 2012 , pp. 27 - 60
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Amiot, C., Cluster categories for algebras of global dimension 2 and quivers with potential, Ann. Inst. Fourier (Grenoble) 59 (6) (2009), 2525–2590.CrossRefGoogle Scholar
2.Assem, I., Brüstle, T., Charbonneau-Jodoin, G. and Plamondon, P., Gentle algebras arising from surface triangulations, Algebra Number Theory 4 (2) (2010), 201–229.CrossRefGoogle Scholar
3.Assem, I. and Dupont, G., Friezes and a construction of the euclidean cluster variables, J. Pure and Appl. Algebra 215 (10) (2011), 2322–2340.Google Scholar
4.Assem, I., Reutenauer, C. and Smith, D., Friezes, Adv. Math. 225 (2010), 3134–3165.CrossRefGoogle Scholar
5.Assem, I., Simson, D. and Skowroński, A., Elements of representation theory of associative algebras, vol. 1: Techniques of representation theory (London Mathematical Society Student Texts, vol. 65) (Cambridge University Press, Cambridge, UK, 2005).Google Scholar
6.Auslander, M., Reiten, I. and Smalø, S., Cambridge studies in advanced mathematics: Vol 36, Representation theory of Artin algebras (Cambridge University Press, Cambridge, UK, 1997). Corrected reprint of the 1995 original.Google Scholar
7.Baur, K. and Marsh, R., Categorification of a frieze pattern determinant. arXiv:1008.5329v1 [math.CO], 2010.Google Scholar
8.Buan, A., Iyama, O., Reiten, I. and Scott, J., Cluster structures for 2-Calabi-Yau categories and unipotent groups, Compos. Math. 145 (4) (2009), 1035–1079.CrossRefGoogle Scholar
9.Buan, A., Iyama, O., Reiten, I. and Smith, D., Mutation of cluster-tilting objects and potentials. To appear in American J. Math. arXiv:0804.3813v3 [math.RT], 2009.Google Scholar
10.Buan, A., Marsh, R., Reineke, M., Reiten, I. and Todorov, G., Tilting theory and cluster combinatorics, Adv. Math. 204 (2) (2006), 572–618.Google Scholar
11.Buan, A., Marsh, R. and Reiten, I., Cluster-tilted algebras, Trans. Amer. Math. Soc. 359 (1) (2007), 323–332.CrossRefGoogle Scholar
12.Buan, A., Marsh, R. and Reiten, I., Denominators of cluster variables, J. Lond. Math. Soc. (2) 79 (3) (2009), 589–611.CrossRefGoogle Scholar
13.Butler, M. and Ringel, C. M., Auslander-Reiten sequences with few middle terms and applications to string algebras, Commun. Algebra, 15 (1–2) (1987), 145–179.CrossRefGoogle Scholar
14.Caldero, P. and Chapoton, F., Cluster algebras as Hall algebras of quiver representations, Commentarii Mathematici Helvetici 81 (2006), 596–616.CrossRefGoogle Scholar
15.Caldero, P., Chapoton, F. and Schiffler, R., Quivers with relations arising from clusters (A n case), Trans. AMS 358 (2006), 1347–1354.Google Scholar
16.Caldero, P. and Keller, B., From triangulated categories to cluster algebras II, Annales Scientifiques de l'Ecole Normale Supérieure, 39 (4) (2006), 83–100.Google Scholar
17.Caroll, G. and Price, G., The new combinatorial models for the Ptolemy recurrence. unpublished.Google Scholar
18.Irelli, G. Cerulli, Quiver Grassmannians associated with string modules, J. Algebr. Comb. 33 (2) (2011), 259–276.CrossRefGoogle Scholar
19.Conway, J. and Coxeter, H., Triangulated polygons and frieze patterns, Math. Gaz. 57 (401) (1973), 175–183.CrossRefGoogle Scholar
20.Conway, J. and Coxeter, H., Triangulated polygons and frieze patterns, Math. Gaz. 57 (400) (1973), 87–94.Google Scholar
21.Coxeter, H., Frieze patterns, Acta Arith. 18 (1971), 297–310.Google Scholar
22.Derksen, H., Weyman, J. and Zelevinsky, A., Quivers with potentials and their representations. I. Mutations, Selecta Math. (N.S.) 14 (1) (2008), 59–119.CrossRefGoogle Scholar
23.Dupont, G., Positivity in coefficient-free rank two cluster algebras, Electron. J. Comb. 16 (1) (2009), R98.Google Scholar
24.Dupont, G., Caldero–Keller approach to the denominators of cluster variables, Commun. Algebra 38 (7) (2010), 2538–2549.Google Scholar
25.Dupont, G., Cluster multiplication in regular components via generalized Chebyshev polynomials, Algebr. Represent. Theory. To appear.Google Scholar
26.Dupont, G., Quantized Chebyshev polynomials and cluster characters with coefficients, J. Algebr. Comb. 31 (4) (2010), 501–532.CrossRefGoogle Scholar
27.Fomin, S., Shapiro, M. and Thurston, D., Cluster algebras and triangulated surfaces. I. Cluster complexes, Acta Math. 201 (1) (2008), 83–146.Google Scholar
28.Fomin, S. and Zelevinsky, A., Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002), 497–529.CrossRefGoogle Scholar
29.Fomin, S. and Zelevinsky, A., Cluster algebras IV: Coefficients, Compos. Math. 143 (1) (2007), 112–164.Google Scholar
30.Fordy, A. and Marsh, R., Cluster mutation-periodic quivers and associated Laurent sequences, J. Algebr. Comb. 34 (1) (2011), 19–66.Google Scholar
31.Fu, C. and Keller, B., On cluster algebras with coefficients and 2-Calabi-Yau categories, Trans. Amer. Math. Soc. 362 (2010), 859–895.Google Scholar
32.Fu, C. and Liu, P., Lifting to cluster-tilting objects in 2-Calabi-Yau triangulated categories, Comm. Algebra 37 (7) (2009), 2410–2418.Google Scholar
33.Gabriel, P., Auslander–Reiten sequences and representation-finite algebras, In Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), vol. 831 of Lecture Notes in Math. (Springer, Berlin, 1980), 1–71.Google Scholar
34.Haupt, N., Euler characteristics of quiver Grassmannians and Ringel–Hall algebras of string algebras, Algebra. Represent. Theory. To appear.Google Scholar
35.Keller, B., On triangulated orbit categories, Doc. Math. 10 (2005), 551–581.CrossRefGoogle Scholar
36.Keller, B. and Van den Bergh, M., Deformed Calabi–Yau completions, J. reine angew. Math. 2011 (654) (2011), 125–180.Google Scholar
37.Keller, B. and Reiten, I., Cluster-tilted algebras are Gorenstein and stably Calabi–Yau, Adv. Math. 211 (1) (2007), 123–151.Google Scholar
38.Keller, B. and Reiten, I., Acyclic Calabi–Yau categories, Compos. Math. 144 (5) (2008), 1332–1348.CrossRefGoogle Scholar
39.Keller, B. and Scherotzke, S., Linear recurrence relations for cluster variables of affine quivers. arXiv:1004.0613v2 [math.RT], 2010.Google Scholar
40.Labardini-Fragoso, D., Quivers with potentials associated to triangulated surfaces, Proc. London Math. Soc. 98 (3) (2009), 787–839.CrossRefGoogle Scholar
41.Morier-Genoud, S., Osvienko, V. and Tabachnikov, S., 2-frieze patterns and the cluster structure of the space of polygons. arXiv:1008.3359v2 [math.AG], 2010.Google Scholar
42.Musiker, G., A graph theoretic expansion formula for cluster algebras of classical type, Ann. Comb. 15 (1) (2011), 147–184.CrossRefGoogle Scholar
43.Musiker, G. and Propp, J., Combinatorial interpretations for rank-two cluster algebras of affine type, Electron. J. Combin. 14 (1) (2007), Research Paper 15, 23 pp. (electronic).Google Scholar
44.Musiker, G., Schiffler, R. and Williams, L., Positivity for cluster algebras from surfaces, Adv. Math. doi:10.1016/j.aim.2011.04.018.Google Scholar
45.Nakajima, H., Quiver varieties and cluster algebras, Kyoto J. Math. 51 (1) (2011), 71–126.CrossRefGoogle Scholar
46.Palu, Y., Cluster characters for 2-Calabi-Yau triangulated categories, Ann. Inst. Fourier (Grenoble) 58 (6) (2008), 2221–2248.CrossRefGoogle Scholar
47.Plamondon, P., Cluster algebras via cluster categories with infinite-dimensional morphism spaces. arXiv:1004.0830v2 [math.RT], 2010.Google Scholar
48.Propp, J., The combinatorics of frieze patterns and Markoff numbers. arXiv:math/0511633v4 [math.CO], 2008.Google Scholar
49.Qin, F., Quantum cluster variables via Serre polynomials. arXiv:1004.4171v2 [math.QA], 2010.Google Scholar
50.Reiten, I., Calabi–Yau categories. Talk at the meeting: Calabi–Yau algebras and N-Koszul algebras (CIRM), 2007.Google Scholar
51.Schiffler, R., On cluster algebras arising from unpunctured surfaces II, Adv. Math. 223 (2010), 1885–1923.CrossRefGoogle Scholar
52.Schiffler, R. and Thomas, H., On cluster algebras arising from unpunctured surfaces, Int. Math. Res. Not. 17 (2009), 3160–3189.Google Scholar
53.Sherman, P. and Zelevinsky, A., Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Mosc. Math. J. 4 (2004), 947–974.CrossRefGoogle Scholar
54.Smith, D., On tilting modules over cluster-tilted algebras, Illinois J. Math. 52 (2008), 1223–1247.Google Scholar
55.Wald, B. and Waschbüsch, J., Tame biserial algebras, J. Algebra 95 (2) (1985) 480–500.Google Scholar