On orthogonal and special orthogonal invariants of a single matrix of small order

TL;DR

This paper investigates the polynomial invariants of traceless matrices under orthogonal group actions, constructing minimal generating sets and decompositions for small matrix sizes, advancing understanding of their algebraic structure.

Contribution

It constructs minimal generating sets and Hironaka decompositions for the invariants of traceless matrices under SO_n and O_n actions for small n, including new results for n=4 and n=5.

Findings

Constructed a minimal generating set for n=5

Developed a Hironaka decomposition for n=3 and 4

Provided a simple presentation with one syzygy for n=3

Abstract

We study the structure of the algebra of polynomial invariants for the usual conjugation action of the complex special, SO_n, and general, O_n, 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.