On orthogonal invariants in characteristic 2

Working over an algebraically closed base field $k$ of characteristic 2, the ring of invariants $R^G$ is studied, where $G$ is the orthogonal group O(n) or the special orthogonal group SO(n), acting naturally on the coordinate ring $R$ of the $m$-fold direct sum $k^n \oplus...\oplus k^n$ of the standard vector representation. It is proved for O(2), $O(3)=SO(3)$, SO(4), and O(4), that there exists an $m$-linear invariant with $m$ arbitrarily large, which is not expressible as a polynomial of invariants of lower degree. This is in sharp contrast with the uniform description of the ring of invariants valid in all other characteristics, and supports the conjecture that the same phenomena occur for all $n$. For general even $n$, new O(n)-invariants are constructed, which are not expressible as polynomials of the quadratic invariants. In contrast with these results, it is shown that rational invariants have a uniform description valid in all characteristics. Similarly, if $m \leq n$, then $R^O(n)$ is generated by the obvious invariants. For all $n$, the algebra $R^G$ is a finitely generated module over the subalgebra generated by the quadratic invariants, and for odd $n$, the square of any SO(n)-invariant is a polynomial of the quadratic invariants. Finally we mention that for even $n$, an $n$-linear SO(n)-invariant is given, which distinguishes between SO(n) and O(n) (just like the determinant in all characteristics different from 2).