On covariants in exterior algebras for the even special orthogonal group

Let G:=SO(2n,C)G:=SO(2n,C) be the even special orthogonal group and let M2n+ (resp. M2n−) be the space of symmetric (resp. skew-symmetric) complex matrices with respect to the usual transposition.We study the structure of B+:=(⋀(M2n+)⁎⊗M2n−)G, the space of G-equivariant skew-symmetric matrix valued alternating multilinear maps on the space of symmetric n-tuples of matrices, with G acting by conjugation.Further, we decompose B   as the direct sum B≃B+⊕B−B≃B+⊕B−, where B∓:=(⋀(M2n•)⁎⊗M2n±)G.We prove that B+B+ is a free module over a certain subalgebra of invariants A:=(⋀(M2n+)⁎)G of rank 2n. We give an explicit description for the basis of this module. Furthermore we prove new trace polynomial identities for symmetric matrices.Finally we show, using a computer assisted computation made with the LiE software, that B−:=(⋀(M2n+)⁎⊗M2n+)G doesn't satisfy a similar property.