On the orbits of an orthogonal group action

Czarnecki, Kyle; Howe, R. Michael; McTavish, Aaron

doi:10.2140/involve.2009.2.495

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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

DOI: 10.2140/involve.2009.2.495

Abstract

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

(g, h) . v : = g^{- 1} v h, (g, h) \in G, 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

Milestones

Received: 8 April 2008

Accepted: 28 September 2009

Published: 13 January 2010

Communicated by Józef H. Przytycki

Authors

Kyle Czarnecki
Department of Mathematics University of Wisconsin – Parkside 900 Wood Rd. P.O. Box 2000 Kenosha, WI 53141-2000 United States

R. Michael Howe
Department of Mathematics University of Wisconsin – Eau Claire 508 Hibbard Humanities Hall Eau Claire, WI 54702-4004 United States
http://www.uwec.edu/math/Faculty/howe.htm
Aaron McTavish
Department of Mathematical Sciences University of Wisconsin – Stevens Point Stevens Point, WI 54481-3897 United States

Vol. 2, No. 5, 2009