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

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.