Mod 2 indecomposable orthogonal invariants

Over an algebraically closed base field k of characteristic 2, the ring RG of invariants is studied, G being the orthogonal group O(n) or the special orthogonal group SO(n) and acting naturally on the coordinate ring R of the m-fold direct sum kn⊕⋯⊕kn of the standard vector representation. It is proved for O(n) (n⩾2) and for SO(n) (n⩾3) that there exist m-linear invariants with m arbitrarily large that are indecomposable (i.e., not expressible as polynomials in invariants of lower degree). In fact, they are explicitly constructed for all possible values of m. Indecomposability of corresponding invariants over Z immediately follows. The constructions rely on analysing the Pfaffian of the skew-symmetric matrix whose entries above the diagonal are the scalar products of the vector variables.