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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:[email protected]
Wayne Whitney
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, U.S.A., e-mail:[email protected]
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Abstract

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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

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