Graphs having no quantum symmetry
Annales de l'Institut Fourier, Volume 57 (2007) no. 3, pp. 955-971.

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 pk the graph has no quantum symmetry, in the sense that the quantum automorphism group reduces to the classical automorphism group.

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 pk 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.

DOI: 10.5802/aif.2282
Classification: 16W30,  05C25,  20B25
Keywords: Quantum permutation group, circulant graph
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)
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Banica, Teodor; Bichon, Julien; Chenevier, Gaëtan. Graphs having no quantum symmetry. Annales de l'Institut Fourier, Volume 57 (2007) no. 3, pp. 955-971. doi : 10.5802/aif.2282. http://archive.numdam.org/articles/10.5802/aif.2282/

[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 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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