Indecomposable orthogonal invariants of several matrices over a field of positive characteristic

TL;DR

This paper studies the algebra of invariants of multiple matrices under orthogonal group actions over fields with positive characteristic, revealing that indecomposable invariants can have unbounded degree depending on the characteristic and matrix type.

Contribution

It establishes that for certain characteristics, the degrees of indecomposable invariants grow without bound as the number of matrices increases, extending known results to orthogonal group actions.

Findings

Indecomposable invariants' degrees tend to infinity for 0<p≤n as d increases.

No uniform bound C(n) exists for degrees of invariants in these cases.

Behavior differs for other characteristics and matrix types, such as symmetric and skew-symmetric matrices.

Abstract

We consider the algebra of invariants of $d$-tuples of $n\times n$ matrices under the action of the orthogonal group by simultaneous conjugation over an infinite field of characteristic $p$ different from two. It is well-known that this algebra is generated by the coefficients of the characteristic polynomial of all products of generic and transpose generic $n\times n$ matrices. We establish that in case $0<p\leq n$ the maximal degree of indecomposable invariants tends to infinity as $d$ tends to infinity. In other words, there does not exist a constant $C(n)$ such that it only depends on $n$ and the considered algebra of invariants is generated by elements of degree less than $C(n)$ for any $d$. This result is well-known in case of the action of the general linear group. On the other hand, for the rest of $p$ the given phenomenon does not hold. We investigate the same problem for the cases of symmetric and skew-symmetric matrices.