 
Summary: Invariant Theory of Special Orthogonal
Groups
Helmer Aslaksen, EngChye Tan and Chenbo Zhu
Abstract
In this paper we study the action of SO.n/ on mtuples of n n matrices
by simultaneous conjugation. We show that the polynomial invariants are
generated by traces and polarized Pfaffians of skewsymmmetric projections.
We also discuss the same problem for other classical groups.
1 Special Orthogonal Groups
Let F be a field of characteristic 0. If A is a skewsymmetric 2k 2k matrix
over F, we denote the Pfaffian of A by pf A. It satisfies det A D pf2
A and
pf.gAgt
/ D det g pf A. For an arbitrary 2k 2k matrix M , we define epf.M / D
pf.M M t
/ to be the Pfaffian of the skewsymmetric projection of M . This is
clearly an SO.2k; F/ invariant. By abuse of notation we will refer to epf as the
Pfaffian, too.
Let W D W.n; m; F/ be the vector space of mtuples of n n matrices
over F on which a group G GL.n; F/ acts by simultaneous conjugation. For
