Grobner bases for the rings of special orthogonal and 2 x 2 matrix invariants

We present a Grobner basis for the ideal of relations among the standard ge nerators of the algebra of invariants of the special orthogonal group actin g 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 x 2 mat rices. In particular, we present in an explicit form a Grobner basis for th e 2 x 2 matrix invariants. Finally we use a Sagbi basis to show that the al gebra of SO2 invariants is a Koszul algebra. (C) 2001 Academic Press.