On orthogonal and special orthogonal invariants of a single matrix of small order
Abstract
We study the structure of the algebra of polynomial invariants for the usual conjugation action of the complex special, SOn, and general, On, orthogonal group on the space of traceless n by n complex matrices. (Note that these two algebras coincide if n is odd.) Minimal generating sets of these algebras are known for n less than 5. We construct one for n=5. We also construct a Hironaka decomposition in the case n=3 and a new (more economical) such decomposition for n=4. A simple presentation (with just one syzygy) is obtained for the algebra in the case n=3.
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