Vol. 2, No. 5, 2009

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On the orbits of an orthogonal group action

Kyle Czarnecki, R. Michael Howe and Aaron McTavish

Vol. 2 (2009), No. 5, 495–509
Abstract

Let $G$ be the Lie group $SO\left(n,ℝ\right)×SO\left(n,ℝ\right)$ and let $V$ be the vector space of $n×n$ real matrices. An action of $G$ on $V$ is given by

$\left(g,h\right).v:={g}^{-1}vh,\phantom{\rule{1em}{0ex}}\left(g,h\right)\in G,\phantom{\rule{1em}{0ex}}v\in V.$

We consider the orbits of this group action and demonstrate a cross-section to the orbits. We then determine the stabilizer for a typical element in this cross-section and completely describe the fundamental group of an orbit of maximal dimension.

Keywords
representation theory, orbit, Lie group, homotopy group, Clifford algebra
Mathematical Subject Classification 2000
Primary: 22C05, 57S15
Secondary: 55Q52