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

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

(g,h).v := g1vh,(g,h) G,v 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.

representation theory, orbit, Lie group, homotopy group, Clifford algebra
Mathematical Subject Classification 2000
Primary: 22C05, 57S15
Secondary: 55Q52
Received: 8 April 2008
Accepted: 28 September 2009
Published: 13 January 2010

Communicated by Józef H. Przytycki
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
Aaron McTavish
Department of Mathematical Sciences
University of Wisconsin – Stevens Point
Stevens Point, WI 54481-3897
United States