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

Abstract

The algebra of invariants of d-tuples of $n\\times n$ matrices under the action of the orthogonal group by simultaneous conjugation is considered over an infinite field of characteristic $p\\geq0$ 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 invariant tends to infinity as $d$ tends to infinity. 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 phenomena does not hold. We investigate the same problem for the case of symmetric and skew-symmetric matrices.