Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T05:49:17.609Z Has data issue: false hasContentIssue false
  • English
  • Français

Closed Symmetric Overgroups of Sn in On

Published online by Cambridge University Press:  20 November 2018

Chi-Kwong Li  and
Wayne Whitney
Chi-Kwong Li
Affiliation:
Department of Mathematics, The College of William and Mary, Williamsburg, VA 23188, U.S.A., e-mail:ckli@cs.wm.edu
Wayne Whitney
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, U.S.A., e-mail:whitney@math.berkeley.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

20B3020F5520H1515A60Permutation invariant normisometryreflection groups
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 39 , Issue 1 , 01 March 1996 , pp. 83 - 94
Copyright
Copyright © Canadian Mathematical Society 1995

References

[Ba] Bannai, E., On some finite subgroups of GL(n,Q), J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20(1973), 319–340.Google Scholar
[BGJ Benson, C. T. and Grove, L. C., Finite Reflection Groups, Springer-Verlag, New York, 1985.Google Scholar
[Bo] Bourbaki, N., Groupes etAlgèbres de Lie, Chap. 4—6, Hermann, Paris, 1968.Google Scholar
[Bu] 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
[DLR] Dokovic, D. Z., Li, C. K. and Rodman, L., Isometries for symmetric gauge functions, Linear and Multilinear Algebra 30(1991), 81–92.Google Scholar
[GLe] Gordon, Y. and Lewis, D. R., Isometries of Diagonally Symmetric Banach Spaces, Israel J. Math. 28 (1977), 45–67.Google Scholar
[GLo] Gordon, Y. and Loewy, R., Uniqueness of (Δ) Bases and Isometries of Banach Spaces, Math. Ann. 241(1979), 159–180.Google Scholar
[LM] Li, C. K. and Mehta, P., Permutation Invariant Norms, Linear Algebra Appl.Google Scholar
[ST] Schneider, H. and Turner, R. E. L., Matrices hermitian for an absolute norm, Linear and Multilinear Algebra 1(1973), 9–31.Google Scholar