Geodesic
Closed Symmetric Overgroups of Sn in On
Canadian mathematical bulletin, Tome 39 (1996) no. 1, pp. 83-94

Voir la notice de l'article provenant de la source Cambridge University Press

A norm on Rn is said to be permutation invariant if its value is preserved under permutation of the coordinates of a vector. The isometry group of such a norm must be closed, contain Sn and —I, and be conjugate to a subgroup of On , the orthogonal group. Motivated by this, we are interested in classifying all closed groups G such that 〈—I,Sn 〉 < G < On . We use the theory of Lie groups to classify all possible infinite groups G, and use the theory of finite reflection groups to classify all possible finite groups G. In keeping with the original motivation, all groups arising are shown to be isometry groups. This completes the work of Gordon and Lewis, who studied the same problem and obtained the results for n ≥ 13.
DOI : 10.4153/CMB-1996-011-7
Mots-clés : 20B30, 20F55, 20H15, 15A60, Permutation invariant norm, isometry, reflection groups 
Li, Chi-Kwong; Whitney, Wayne. Closed Symmetric Overgroups of Sn in On. Canadian mathematical bulletin, Tome 39 (1996) no. 1, pp. 83-94. doi: 10.4153/CMB-1996-011-7
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Bannai, E., On some finite subgroups of GL(n,Q), J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20(1973), 319–340. Google Scholar

Benson, C. T. and Grove, L. C., Finite Reflection Groups, Springer-Verlag, New York, 1985. Google Scholar

Bourbaki, N., Groupes etAlgèbres de Lie, Chap. 4—6, Hermann, Paris, 1968. Google Scholar

Burnside, W., The determination of all groups of rational linear groups of finite order which contain the symmetric group in the variables, Proc. London Math. Soc. (Series 2) 10(1912), 284—308. Google Scholar

Dokovic, D. Z., Li, C. K. and Rodman, L., Isometries for symmetric gauge functions, Linear and Multilinear Algebra 30(1991), 81–92. Google Scholar

Gordon, Y. and Lewis, D. R., Isometries of Diagonally Symmetric Banach Spaces, Israel J. Math. 28 (1977), 45–67. Google Scholar

Gordon, Y. and Loewy, R., Uniqueness of (Δ) Bases and Isometries of Banach Spaces, Math. Ann. 241(1979), 159–180. Google Scholar

Li, C. K. and Mehta, P., Permutation Invariant Norms, Linear Algebra Appl. Google Scholar

Schneider, H. and Turner, R. E. L., Matrices hermitian for an absolute norm, Linear and Multilinear Algebra 1(1973), 9–31. Google Scholar

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