Let d be a positive integer andFbe a field of characteristic 0. Suppose that for each positive integern,Inis a polynomial invariant of the usual action ofGLn(F)on #x39B;d(Fn), such that fort#x3F5; #x39B;d(Fk) and s #x3F5; #x39B;d(Fl),Ik + l(tls) =Ik(t)It(s), wheret#x22A5;sis defined in #xA7;1.4. Then we say that{In}is an additive family of invariants of the skewsymmetric tensors of degreed, or, briefly, an additive family of invariants. If not all theInare constant we say that the family is non-trivial. We show that in each even degreedthere is a non-trivial additive family of invariants, but that this is not so for any oddd.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.
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