# Characteristic free description of semi-invariants of $2\times 2$ matrices

## TL;DR

This paper provides a minimal generating system for the algebra of semi-invariants of tuples of 2x2 matrices over various fields and rings, and also identifies a characteristic-independent separating system.

## Contribution

It introduces a characteristic-free minimal generating system for semi-invariants of 2x2 matrices and relates it to vector invariants of the special orthogonal group.

## Key findings

- Minimal homogeneous generating system for semi-invariants over characteristic two and integers.
- A characteristic-independent separating system of semi-invariants.
- Connection to vector invariants of the special orthogonal group.

## Abstract

A minimal homogeneous generating system of the algebra of semi-invariants of tuples of two-by-two matrices over an infinite field of characteristic two or over the ring of integers is given. In an alternative interpretation this yields a minimal system of homogeneous generators for the vector invariants of the special orthogonal group of degree four over a field of characteristic two or over the ring of integers. An irredundant separating system of semi-invariants of tuples of two-by-two matrices is also determined, it turns out to be independent of the characteristic.