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

Abstract

We consider the algebra of invariants of [Formula: see text]-tuples of [Formula: see text] matrices under the action of the orthogonal group by simultaneous conjugation over an infinite field of characteristic [Formula: see text] 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 [Formula: see text] matrices. We establish that in case [Formula: see text] the maximal degree of indecomposable invariants tends to infinity as [Formula: see text] tends to infinity. In other words, there does not exist a constant [Formula: see text] such that it only depends on [Formula: see text] and the considered algebra of invariants is generated by elements of degree less than [Formula: see text] for any [Formula: see text]. This result is well-known in case of the action of the general linear group. On the other hand, for the rest of [Formula: see text] the given phenomenon does not hold. We investigate the same problem for the cases of symmetric and skew-symmetric matrices.