On certain modules of covariants in exterior algebras

2014/04/10 by Dolce, Salvatore
#FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1404.2855

Abstract

We study the structure of the space of covariants B:=(\bigwedge (\mathfrak g/\mathfrak k)^*⊗ \mathfrak g)^{\mathfrak k}, for a certain class of infinitesimal symmetric spaces (\mathfrak g,\mathfrak k) such that the space of invariants A:=(\bigwedge (\mathfrak g/\mathfrak k)^*)^{\mathfrak k} is an exterior algebra \wedge (x₁,...,x_r), with r=rk(\mathfrak g)-rk(\mathfrak k). We prove that they are free modules over the subalgebra A_r-1=\wedge (x₁,...,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 (\bigwedge (M_n^±)^*⊗ M_n)^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.

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