Gröbner Bases for the Rings of Special Orthogonal and 2 × 2 Matrix Invariants,

We present a Gröbner basis for the ideal of relations among the standard generators of the algebra of invariants of the special orthogonal group acting on k-tuples of vectors. The cases of SO3 and SO4 are interpreted in terms of the algebras of invariants and semi-invariants of k-tuples of 2 × 2 matrices. In particular, we present in an explicit form a Gröbner basis for the 2 × 2 matrix invariants. Finally we use a Sagbi basis to show that the algebra of SO2 invariants is a Koszul algebra.