Mod 2 indecomposable orthogonal invariants

Abstract Over an algebraically closed base field k of characteristic 2, the ring R G 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 k n ⊕⋯⊕ k n 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.