Abstract
In this paper we study the action of SO(n) on ra-tuples ofnxn 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 characteristic 0. If A is a skewsymmetric 2k x 2k matrix over F, we denote the Pfaffian of A by pf A. It satisfies det^l = pf2 A and pί(gAgt) = detgpf A. For an arbitrary 2k x 2k matrix M, we define pf (M) = pf (M — Mι) to be the Pfaffian of the skewsymmetric projection of M. This is clearly an SO(2A:,F) invariant. By abuse of notation we will refer to pf as the Pfaffian, too. Let W — W(n^ m, F) be the vector space of m-tuples ofnxn matrices over F on which a group G C GL(n, F) acts by simultaneous conjugation. For G = SO(2, F),the invariants P[W(2, m, F)]G were determined in [1]. They are generated by the invariants tτ P(A, A1) and pΐP(A1At) where A G W(2,m,F) and P is noncommutative polynomial.
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