Frieze patterns for punctured discs
Karin Baur
and Robert J. Marsh
DOI: 10.1007/s10801-008-0161-0
Abstract
We construct frieze patterns of type D N with entries which are numbers of matchings between vertices and triangles of corresponding triangulations of a punctured disc. For triangulations corresponding to orientations of the Dynkin diagram of type D N , we show that the numbers in the pattern can be interpreted as specialisations of cluster variables in the corresponding Fomin-Zelevinsky cluster algebra. This is generalised to arbitrary triangulations in an appendix by Hugh Thomas.
Pages: 349–379
Keywords: keywords cluster algebra; frieze pattern; Ptolemy rule; exchange relation; matching; Riemann surface; disc; triangulation
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References
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2. Buan, A.B., Marsh, R.J., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204, 572-618 (2006)
3. Burman, Y.M.: Triangulations of disc with boundary and the homotopy principle for functions without critical points. Ann. Glob. Anal. Geom. 17, 221-238 (1999)
4. Caldero, P.: Algèbres cluster et catégories cluster, talks at the meeting on Algèbres Clusters, Luminy, France, May 2005
5. Caldero, P., Chapoton, F.: Cluster algebras as Hall algebras of quiver representations. Commun. Math. Helv. 81, 595-616 (2006)
6. Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (An case). Trans. Am. Math. Soc. 358, 1347-1364 (2006)
7. Carroll, G., Price, G.: Two new combinatorial models for the Ptolemy recurrence. Unpublished memo (2003)
8. Conway, J.H., Coxeter, H.S.M.: Triangulated discs and frieze patterns. Math. Gaz. 57, 87-94 (1973)
9. Conway, J.H., Coxeter, H.S.M.: Triangulated discs and frieze patterns. Math. Gaz. 57, 175-186 (1973)
10. Fomin, S., Shapiro, D., Thurston, D.: Cluster algebras and triangulated discs. Part I: Cluster complexes. Preprint (2006)
11. Fomin, S., Zelevinsky, A.: Cluster algebras I: Foundations. J. Am. Math. Soc. 15(2), 497-529 (2002)
12. Fomin, S., Zelevinsky, A.: Cluster algebras II: Finite type classification. Invent. Math. 154(1), 63-121 (2003)
13. Fomin, S., Zelevinsky, A.: Y -systems and generalized associahedra. Ann. Math. 158(3), 977-1018 (2003)
14. Kuo, E.: Applications of graphical condensation for enumerating matchings and tilings. Theor. Comput. Sci. 319, 29-57 (2004)
15. Musiker, G.: A graph theoretic expansion formula for cluster algebras of type Bn and Dn. Preprint [math.CO] (2007)
16. Propp, J.: The combinatorics of frieze patterns and Markoff numbers. Preprint (2005)
17. Ringel, C.M.: Tame Algebras and Integral Quadratic Forms. Lecture Notes in Mathematics, vol. 1099.