Generating O(n) with reflections

Morris Leroy Eaton and Michael David Perlman

Vol. 73 (1977), No. 1, 73–80

DOI: 10.2140/pjm.1977.73.73

Abstract

For r ∈ C_n ≡{x|x ∈ Rⁿ,∥x∥ = 1}, let S_r = I_n − 2rr′ where r is a column vector. O(n) denotes the orthogonal group on Rⁿ. If R ⊆ C_n, let ℛ = {S_r|r ∈ R} and let G be the smallest closed subgroup of O(n) which contains ℛ. G is reducible if there is a nontrivial subspace M ⊆ Rⁿ such that gM ⊆ M for all g ∈ G. Otherwise, G is irreducible.

Theorem. If G is infinite and irreducible, then G = O(n).

Mathematical Subject Classification 2000

Primary: 20H20

Milestones

Received: 11 February 1977

Revised: 28 July 1977

Published: 1 November 1977