Graphs having no quantum symmetry
[Graphes n’ayant pas de symétrie quantique]
Banica, Teodor ; Bichon, Julien ; Chenevier, Gaëtan
Annales de l'Institut Fourier, Tome 57 (2007), p. 955-971 / Harvested from Numdam
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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.

Publié le : 2007-01-01
DOI : https://doi.org/10.5802/aif.2282
Classification:  16W30,  05C25,  20B25
Mots clés: groupe quantique de permutation, graphe circulant 
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     author = {Banica, Teodor and Bichon, Julien and Chenevier, Ga\"etan},
     title = {Graphs having no quantum symmetry},
     journal = {Annales de l'Institut Fourier},
     volume = {57},
     year = {2007},
     pages = {955-971},
     doi = {10.5802/aif.2282},
     zbl = {pre05176611},
     mrnumber = {2336835},
     zbl = {1178.05047},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2007__57_3_955_0}
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Banica, Teodor; Bichon, Julien; Chenevier, Gaëtan. Graphs having no quantum symmetry. Annales de l'Institut Fourier, Tome 57 (2007) pp. 955-971. doi : 10.5802/aif.2282. http://gdmltest.u-ga.fr/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