Graphs having no quantum symmetry
Annales de l'Institut Fourier, Volume 57 (2007) no. 3, p. 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 : https://doi.org/10.5802/aif.2282
Classification:  16W30,  05C25,  20B25
Keywords: Quantum permutation group, circulant graph
@article{AIF_2007__57_3_955_0,
     author = {Banica, Teodor and Bichon, Julien and Chenevier, Ga\"etan},
     title = {Graphs having no quantum symmetry},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {3},
     year = {2007},
     pages = {955-971},
     doi = {10.5802/aif.2282},
     zbl = {1178.05047},
     mrnumber = {2336835},
     zbl = {pre05176611},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2007__57_3_955_0}
}
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://www.numdam.org/item/AIF_2007__57_3_955_0/

[1] Alspach, B. Point-symmetric graphs and digraphs of prime order and transitive permutation groups of prime degree,, J. Combinatorial Theory Ser. B, Tome 15 (1973), pp. 12-17 | Article | MR 332553 | Zbl 0271.05117

[2] Banica, T. Symmetries of a generic coaction, Math. Ann., Tome 314 (1999), pp. 763-780 | Article | MR 1709109 | Zbl 0928.46038

[3] Banica, T. Quantum automorphism groups of homogeneous graphs, J. Funct. Anal., Tome 224 (2005), pp. 243-280 | Article | MR 2146039 | Zbl 02189266

[4] Banica, T. Quantum automorphism groups of small metric spaces, Pacific J. Math., Tome 219 (2005), pp. 27-51 | Article | MR 2174219 | Zbl 05011504

[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 1129.46058

[8] Biane, P. Representations of symmetric groups and free probability, Adv. Math., Tome 138 (1998), pp. 126-181 | Article | MR 1644993 | Zbl 0927.20008

[9] Bichon, J. Quantum automorphism groups of finite graphs, Proc. Amer. Math. Soc., Tome 131 (2003), pp. 665-673 | Article | MR 1937403 | Zbl 1013.16032

[10] Bichon, J. Free wreath product by the quantum permutation group, Alg. Rep. Theory, Tome 7 (2004), pp. 343-362 | Article | MR 2096666 | Zbl 02128533

[11] Bisch, D.; Jones, V. F. R. Singly generated planar algebras of small dimension, Duke Math. J., Tome 101 (2000), pp. 41-75 | Article | MR 1733737 | Zbl 1075.46053

[12] Collins, B. Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability, Int. Math. Res. Not., Tome 17 (2003), pp. 953-982 | Article | MR 1959915 | Zbl 1049.60091

[13] Di Francesco, P. Meander determinants, Comm. Math. Phys., Tome 191 (1998), pp. 543-583 | Article | MR 1608551 | Zbl 0923.57002

[14] Dobson, E.; Morris, J. On automorphism groups of circulant digraphs of square-free order, Discrete Math., Tome 299 (2005), pp. 79-98 | Article | MR 2168697 | Zbl 1073.05034

[15] Jones, V. F. R.; Sunder, V. S. Introduction to subfactors, LMS Lecture Notes, Cambridge University Press, Tome 234 (1997) | MR 1473221 | Zbl 0903.46062

[16] Klimyk, A.; Schmüdgen, K. Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin (1997) | MR 1492989 | Zbl 0891.17010

[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, Tome 25 (1981), pp. 405-434 | MR 642055 | Zbl 0478.05046

[18] Wang, S. Quantum symmetry groups of finite spaces, Comm. Math. Phys., Tome 195 (1998), pp. 195-211 | Article | MR 1637425 | Zbl 1013.17008

[19] Washington, L. C. Introduction to cyclotomic fields, GTM, Springer, Tome 83 (1982) | MR 718674 | Zbl 0484.12001

[20] Weingarten, D. Asymptotic behavior of group integrals in the limit of infinite rank, J. Math. Phys., Tome 19 (1978), pp. 999-1001 | Article | MR 471696 | Zbl 0388.28013

[21] Woronowicz, S.L. Compact matrix pseudogroups, Comm. Math. Phys., Tome 111 (1987), pp. 613-665 | Article | MR 901157 | Zbl 0627.58034