Symplectic polynomial invariants of one or two matrices of small size

TL;DR

This paper constructs explicit minimal generating sets and decompositions for the algebra of symplectic polynomial invariants of small-sized matrices, advancing understanding of these algebraic structures.

Contribution

It provides explicit minimal generating sets and a Hironaka decomposition for the algebra of symplectic invariants in specific small cases, which were previously unknown.

Findings

Constructed minimal generating sets for n=k=2 and n=3, k=1 cases.

Established a homogeneous system of parameters for the algebra.

Provided a Hironaka decomposition in the n=3, k=1 case.

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.