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.

中文翻译：

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若干矩阵在正特征场上的不可分解正交不变量

我们考虑在正交群的作用下 [Formula: see text]-tuples of [Formula: see text] 矩阵的不变量的代数，通过同时共轭在特征 [Formula: see text] 的无限域上进行，而不是两个。众所周知，这个代数是由泛型和转置泛型[公式：见正文]矩阵的所有乘积的特征多项式的系数生成的。我们确定在 [公式：见文本] 的情况下，不可分解不变量的最大程度趋于无穷大，因为 [公式：见文本] 趋于无穷大。换句话说，不存在一个常数 [Formula: see text]，它只依赖于 [Formula: see text] 并且所考虑的不变量代数是由任何程度小于 [Formula: see text] 的元素生成的[公式：见正文]。这个结果在一般线性群的作用下是众所周知的。另一方面，对于[公式：见正文]的其余部分，给定现象不成立。我们针对对称矩阵和斜对称矩阵的情况研究了相同的问题。