On covariants in exterior algebras for the even special orthogonal group

Abstract

Let $G:=SO(2n)$ be the even special orthogonal group over $\mathbb{C}$ and let $M_{2n}^+$ (resp. $M_{2n}^-$) be the space of symmetric (resp. skew-symmetric) complex matrices with respect to the usual transposition. We study the structure of the space $B^+:=\left(\bigwedge (M_{2n}^{+})^*\otimes M_{2n}^-\right)^G$, 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. We prove that $B^+$ is a free module over a certain subalgebra of invariants $A:=\left(\bigwedge (M_{2n}^{+})^*\right)^G$ 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 computation made with the LiE software, that the analogous module $B^-:=\left(\bigwedge (M_{2n}^{+})^*\otimes M_{2n}^+\right)^G$ doesn't satisfy a similar property.