On certain modules of covariants in exterior algebras

Abstract

We study the structure of the space of covariants $B:={\left(\bigwedge (\mathfrak{g}/\mathfrak{k}{)}^{\ast }\otimes \mathfrak{g}\right)}^{\mathfrak{k}},$ for a certain class of infinitesimal symmetric spaces $(\mathfrak{g},\mathfrak{k})$ such that the space of invariants $A:={\left(\bigwedge (\mathfrak{g}/\mathfrak{k}{)}^{\ast }\right)}^{\mathfrak{k}}$ is an exterior algebra $\wedge (x_1,...,x_r),$ with $r=rk(\mathfrak{g})-rk(\mathfrak{k})$. We prove that they are free modules over the subalgebra $A_{r-1}=\wedge (x_1,...,x_{r-1})$ of rank $4r$. In addition we will give an explicit basis of $B$. As particular cases we will recover same classical results. In fact we will describe the structure of ${\left(\bigwedge (M_n^{\pm }{)}^{\ast }\otimes M_n\right)}^G$, the space of the $G-$equivariant matrix valued alternating multilinear maps on the space of (skew-symmetric or symmetric with respect to a specific involution) matrices, where $G$ is the symplectic group or the odd orthogonal group. Furthermore we prove new polynomial trace identities.

Title changed. Results have been generalised to other infinitesimal symmetric spaces