Symplectic polynomial invariants of one or two matrices of small size

Abstract

The algebra of holomorphic polynomial Sp_{2n}-invariants of k complex 2n by 2n matrices (under diagonal conjugation action) is generated by the traces of words in these matrices and their symplectic adjoints. No concrete minimal generating set is known for this algebra apart from the cases n=1, when Sp_2=SL_2, and n=2, k=1. We construct such sets in the cases n=k=2 and n=3, k=1. In the latter case we also construct a homogeneous system of parameters and a Hironaka decomposition of the algebra.