Graphs having no quantum symmetry

Banica, Teodor; Bichon, Julien; Chenevier, Gaëtan

doi:10.5802/aif.2282

Graphs having no quantum symmetry
[Graphes n’ayant pas de symétrie quantique]

Banica, Teodor ¹ ; Bichon, Julien ² ; Chenevier, Gaëtan ³

¹ Université Toulouse 3 Département de mathématiques 118, route de Narbonne 31062 Toulouse (France)
² Université de Pau Département de mathématiques 1, avenue de l’université 64000 Pau (France)
³ Université Paris 13 Département de mathématiques 99, avenue J-B. Clément 93430 Villetaneuse (France)

Annales de l'Institut Fourier, Tome 57 (2007) no. 3, pp. 955-971.

Résumé
Abstract

On considère des graphes circulants ayant $p$ sommets, avec $p$ premier. A un tel graphe on associe un certain nombre $k$ , qu’on appelle type du graphe. On montre que pour $p ≫ k$ le graphe n’a pas de symétrie quantique, dans le sens où son groupe quantique d’automorphismes est réduit à son groupe classique d’automorphismes.

We consider circulant graphs having $p$ vertices, with $p$ prime. To any such graph we associate a certain number $k$ , that we call type of the graph. We prove that for $p ≫ k$ the graph has no quantum symmetry, in the sense that the quantum automorphism group reduces to the classical automorphism group.

EuDML MR Zbl

DOI : 10.5802/aif.2282

Classification : 16W30, 05C25, 20B25
Keywords: Quantum permutation group, circulant graph
Mot clés : groupe quantique de permutation, graphe circulant

Affiliations des auteurs :

Banica, Teodor ¹ ; Bichon, Julien ² ; Chenevier, Gaëtan ³

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     title = {Graphs having no quantum symmetry},
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Banica, Teodor; Bichon, Julien; Chenevier, Gaëtan. Graphs having no quantum symmetry. Annales de l'Institut Fourier, Tome 57 (2007) no. 3, pp. 955-971. doi : 10.5802/aif.2282. http://archive.numdam.org/articles/10.5802/aif.2282/

Bibliographie
Cité par

[1] Alspach, B. Point-symmetric graphs and digraphs of prime order and transitive permutation groups of prime degree,, J. Combinatorial Theory Ser. B, Volume 15 (1973), pp. 12-17 | DOI | MR | Zbl

[2] Banica, T. Symmetries of a generic coaction, Math. Ann., Volume 314 (1999), pp. 763-780 | DOI | MR | Zbl

[3] Banica, T. Quantum automorphism groups of homogeneous graphs, J. Funct. Anal., Volume 224 (2005), pp. 243-280 | DOI | MR | Zbl

[4] Banica, T. Quantum automorphism groups of small metric spaces, Pacific J. Math., Volume 219 (2005), pp. 27-51 | DOI | MR | Zbl

[5] Banica, T.; Bichon, J. Free product formulae for quantum permutation groups (J. Math. Inst. Jussieu, to appear)

[6] Banica, T.; Bichon, J. Quantum automorphism groups of vertex-transitive graphs of order $\leq$ 11 (math.QA/0601758)

[7] Banica, T.; Collins, B. Integration over compact quantum groups (Publ. Res. Inst. Math. Sci., to appear) | Zbl

[8] Biane, P. Representations of symmetric groups and free probability, Adv. Math., Volume 138 (1998), pp. 126-181 | DOI | MR | Zbl

[9] Bichon, J. Quantum automorphism groups of finite graphs, Proc. Amer. Math. Soc., Volume 131 (2003), pp. 665-673 | DOI | MR | Zbl

[10] Bichon, J. Free wreath product by the quantum permutation group, Alg. Rep. Theory, Volume 7 (2004), pp. 343-362 | DOI | MR | Zbl

[11] Bisch, D.; Jones, V. F. R. Singly generated planar algebras of small dimension, Duke Math. J., Volume 101 (2000), pp. 41-75 | DOI | MR | Zbl

[12] Collins, B. Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability, Int. Math. Res. Not., Volume 17 (2003), pp. 953-982 | DOI | MR | Zbl

[13] Di Francesco, P. Meander determinants, Comm. Math. Phys., Volume 191 (1998), pp. 543-583 | DOI | MR | Zbl

[14] Dobson, E.; Morris, J. On automorphism groups of circulant digraphs of square-free order, Discrete Math., Volume 299 (2005), pp. 79-98 | DOI | MR | Zbl

[15] Jones, V. F. R.; Sunder, V. S. Introduction to subfactors, LMS Lecture Notes, Volume 234, Cambridge University Press, 1997 | MR | Zbl

[16] Klimyk, A.; Schmüdgen, K. Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997 | MR | Zbl

[17] Klin, M. H.; Pöschel, R. The König problem, the isomorphism problem for cyclic graphs and the method of Schur rings,, Colloq. Math. Soc. Janos Bolyai, Volume 25 (1981), pp. 405-434 | MR | Zbl

[18] Wang, S. Quantum symmetry groups of finite spaces, Comm. Math. Phys., Volume 195 (1998), pp. 195-211 | DOI | MR | Zbl

[19] Washington, L. C. Introduction to cyclotomic fields, GTM, Volume 83, Springer, 1982 | MR | Zbl

[20] Weingarten, D. Asymptotic behavior of group integrals in the limit of infinite rank, J. Math. Phys., Volume 19 (1978), pp. 999-1001 | DOI | MR | Zbl

[21] Woronowicz, S.L. Compact matrix pseudogroups, Comm. Math. Phys., Volume 111 (1987), pp. 613-665 | DOI | MR | Zbl

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