 
Summary: PACIFIC JOURNAL OF MATHEMATICS
Vol. 168, No. 2, 1995
INVARIANT THEORY OF SPECIAL ORTHOGONAL
GROUPS
HELMER ASLAKSEN, ENGCHYE TAN AND CHENBO ZHU
In this paper we study the action of SO(n) on ratuples
o f n x n matrices by simultaneous conjugation. We show
that the polynomial invariants are generated by traces
and polarized Pfaffians of skewsymmetric projections. We
also discuss the same problem for other classical groups.
1. Special orthogonal groups. Let F be a field of charac
teristic 0. If A is a skewsymmetric 2k x 2k matrix over F, we
denote thePfaffian of A by pfA. It satisfies det^l = pf2
Aand
p(gAgt
) = detgpf A. For an arbitrary 2k x 2k matrix M, we de
fine pf (M) = pf (M M
) tobe the Pfaffian of the skewsymmetric
projection of M. This is clearly anSO(2A:,F) invariant. By abuse
of notation wewill refer topf as the Pfaffian, too.
