Counting Coxeter’s friezes over a finite field via moduli spaces

Morier-Genoud, Sophie

doi:10.5802/alco.140

Morier-Genoud, Sophie ¹

¹ Sorbonne Université Université de Paris, CNRS Institut de Mathématiques de Jussieu-Paris Rive Gauche IMJ-PRG F-75005, Paris, France,

Algebraic Combinatorics, Tome 4 (2021) no. 2, pp. 225-240.

Résumé

We count the number of Coxeter’s friezes over a finite field. Our method uses geometric realizations of the spaces of friezes in a certain completion of the classical moduli space $ℳ_{0, n}$ allowing repeated points in the configurations. Counting points in the completed moduli space over a finite field is related to the enumeration problem of counting partitions of cyclically ordered set of points into subsets containing no consecutive points. In the appendix we provide an elementary solution for this enumeration problem.

Reçu le : 2019-07-30
Révisé le : 2020-07-17
Accepté le : 2020-07-27
Publié le : 2021-04-12

DOI : 10.5802/alco.140

Classification : 13F60, 11G25, 05A18
Mots clés : Frieze, Moduli space, Finite field, Partitions, Stirling numbers, cluster variety

Affiliations des auteurs :

Morier-Genoud, Sophie ¹

¹ Sorbonne Université Université de Paris, CNRS Institut de Mathématiques de Jussieu-Paris Rive Gauche IMJ-PRG F-75005, Paris, France,

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Morier-Genoud, Sophie. Counting Coxeter’s friezes over a finite field via moduli spaces. Algebraic Combinatorics, Tome 4 (2021) no. 2, pp. 225-240. doi : 10.5802/alco.140. http://www.numdam.org/articles/10.5802/alco.140/

Bibliographie
Cité par

[1] Baur, Karin; Marsh, Robert J. Frieze patterns for punctured discs, J. Algebraic Combin., Volume 30 (2009) no. 3, pp. 349-379 | DOI | MR | Zbl

[2] Bergeron, François; Reutenauer, Christophe $S L_{k}$ -tilings of the plane, Illinois J. Math., Volume 54 (2010) no. 1, pp. 263-300 | DOI | MR | Zbl

[3] Boalch, Philip Wild character varieties, points on the Riemann sphere and Calabi’s examples, Representation theory, special functions and Painlevé equations—RIMS 2015 (Adv. Stud. Pure Math.), Volume 76, Math. Soc. Japan, Tokyo, 2018, pp. 67-94 | DOI | MR

[4] 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 | Zbl

[5] Chapoton, Frédéric On the number of points over finite fields on varieties related to cluster algebras, Glasg. Math. J., Volume 53 (2011) no. 1, pp. 141-151 | DOI | MR | Zbl

[6] Chapoton, Frédéric On some varieties associated with trees, Michigan Math. J., Volume 64 (2015) no. 4, pp. 721-758 | DOI | MR | Zbl

[7] Conway, John H.; Coxeter, Harold S. M. Triangulated polygons and frieze patterns, Math. Gaz., Volume 57 (1973) no. 400, 401, p. 87-94, 175–183 | DOI | MR | Zbl

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

[9] Coxeter, Harold S. M. Regular complex polytopes, Cambridge University Press, Cambridge, 1991, xiv+210 pages | MR | Zbl

[10] Cuntz, Michael On wild frieze patterns, Exp. Math., Volume 26 (2017) no. 3, pp. 342-348 | DOI | MR | Zbl

[11] Cuntz, Michael A combinatorial model for tame frieze patterns, Münster J. Math., Volume 12 (2019) no. 1, pp. 49-56 | DOI | MR | Zbl

[12] Cuntz, Michael; Holm, Thorsten Frieze patterns over integers and other subsets of the complex numbers, J. Comb. Algebra, Volume 3 (2019) no. 2, pp. 153-188 | DOI | MR | Zbl

[13] Fontaine, Bruce Non-zero integral friezes (2014) (https://arxiv.org/abs/1409.6026)

[14] Fontaine, Bruce; Plamondon, Pierre-Guy Counting friezes in type $D_{n}$ , J. Algebraic Combin., Volume 44 (2016) no. 2, pp. 433-445 | DOI | MR | Zbl

[15] Galvin, David; Thanh, Do Trong Stirling numbers of forests and cycles, Electron. J. Combin., Volume 20 (2013) no. 1, 73, 16 pages | MR | Zbl

[16] Gelʼfand, Israel M.; MacPherson, Robert D. Geometry in Grassmannians and a generalization of the dilogarithm, Adv. in Math., Volume 44 (1982) no. 3, pp. 279-312 | DOI | MR | Zbl

[17] Holm, Thorsten; Jørgensen, Peter A $p$ -angulated generalisation of Conway and Coxeter’s theorem on frieze patterns, Int. Math. Res. Not. IMRN (2020) no. 1, pp. 71-90 | DOI | MR

[18] Knuth, Donald E.; Lossers, O. P. Partitions of a circular set. Problem 11151, Volume 114, 2007 no. 3, pp. 265-266

[19] Mabilat, Flavien Combinatorial description of the principal congruence subgroups $Γ (2)$ in $S L (2, ℤ)$ (2019) (https://arxiv.org/abs/1911.06717)

[20] Morier-Genoud, Sophie Coxeter’s frieze patterns at the crossroads of algebra, geometry and combinatorics, Bull. Lond. Math. Soc., Volume 47 (2015) no. 6, pp. 895-938 | DOI | MR | Zbl

[21] Morier-Genoud, Sophie; Ovsienko, Valentin; Schwartz, Richard E.; Tabachnikov, Serge Linear difference equations, frieze patterns, and the combinatorial Gale transform, Forum Math. Sigma, Volume 2 (2014), e22, 45 pages | DOI | MR | Zbl

[22] Morier-Genoud, Sophie; Ovsienko, Valentin; Tabachnikov, Serge 2-frieze patterns and the cluster structure of the space of polygons, Ann. Inst. Fourier, Volume 62 (2012) no. 3, pp. 937-987 | DOI | Numdam | MR | Zbl

[23] OEIS, OEIS Foundation Inc The On-Line Encyclopedia of Integer Sequences, 2019 (http://oeis.org)

[24] Ovsienko, Valentin Partitions of unity in $S L (2, ℤ)$ , negative continued fractions, and dissections of polygons, Res. Math. Sci., Volume 5 (2018) no. 2, 21, 25 pages | DOI | MR | Zbl

[25] Sibuya, Yasutaka Global theory of a second order linear ordinary differential equation with a polynomial coefficient, North-Holland Mathematics Studies, 18, North-Holland Publishing Co., Amsterdam-Oxford; Elsevier Publishing Co., Inc., New York, 1975, xv+290 pages | MR | Zbl

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