On covariants in exterior algebras for the even special orthogonal group

TL;DR

This paper investigates the structure of covariant spaces in exterior algebras related to the even special orthogonal group, revealing freeness over invariants, explicit bases, and new trace identities, with contrasting results for related modules.

Contribution

It provides a detailed structural analysis of covariant modules in exterior algebras for SO(2n), including freeness results, explicit bases, and new trace polynomial identities, extending understanding of invariants in this setting.

Findings

B^+ is a free module over a subalgebra of invariants with rank 2n.

Explicit basis for the free module B^+ is constructed.

New trace polynomial identities for symmetric matrices are established.

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.