Relations between O(n)-invariants of several matrices

TL;DR

This paper explores the algebraic relations between invariants of matrices under the action of the orthogonal group O(n), extending classical results to fields of positive characteristic.

Contribution

It generalizes known results on matrix invariants from characteristic zero to positive characteristic, focusing on relations for O(n)-invariants.

Findings

Established relations between generators for O(n)-invariants in positive characteristic

Extended classical invariant theory results to new algebraic settings

Provided a framework for understanding matrix invariants under orthogonal group actions

Abstract

A linear group G<GL(n) acts on d-tuples of n x n matrices by simultaneous conjugation. In [Adv. Math. 19 (1976), 306-381] Procesi established generators and relations between them for G-invariants, where G is GL(n), O(n), and Sp(n) and the characteristic of base field is zero. We continue generalization of the mentioned results to the case of positive characteristic originated by Donkin in [Invent. Math. 110 (1992), 389-401]. We investigate relations between generators for O(n)-invariants.