| Abstract: | Abstract Let d be a positive integer and F be a field of characteristic 0. Suppose that for each positive integer n, I n is a polynomial invariant of the usual action of GLn (F) on Λd(Fn), such that for t ? Λd(F k) and s ? Λd(F l), I k + l (t l s) = I k(t)I t (s), where t ⊥ s is defined in §1.4. Then we say that {In} is an additive family of invariants of the skewsymmetric tensors of degree d, or, briefly, an additive family of invariants. If not all the In are constant we say that the family is non-trivial. We show that in each even degree d there is a non-trivial additive family of invariants, but that this is not so for any odd d. These results are analogous to those in our paper [3] for symmetric tensors. Our proofs rely on the symbolic method for representing invariants of skewsymmetric tensors. To keep this paper self-contained we expound some of that theory, but for the proofs we refer to the book [2] of Grosshans, Rota and Stein. |
