On covariants in exterior algebras for the even special orthogonal group

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• 作者：Salvatore Dolce^a ; ^b ; ^{gsdolce@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Invariant theory ; Polynomial identities ; Exterior algebras ; Algebras with involutions ; Symmetric matrices
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 September 2016
• 年：2016
• 卷：462
• 期：Complete
• 页码：163-180
• 全文大小：372 K

文摘

Let G:=SO(2n,C)$G:=SO(2n,\mathbb{C})$ be the even special orthogonal group 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 $B^{+}:={(\bigwedge {(M_{2n}^{+})}^{⁎}\otimes M_{2n}^{-})}^G$, 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⁻$B\simeq B^{+}\oplus B^{-}$, where $B^{\mp }:={(\bigwedge {(M_{2n}^{•})}^{⁎}\otimes M_{2n}^{\pm })}^G$.

We prove that 2013b863882958c640d7fe4f75563e6" title="Click to view the MathML source">B⁺$B^{+}$ is a free module over a certain subalgebra of invariants $A:={(\bigwedge {(M_{2n}^{+})}^{⁎})}^G$ 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 $B^{-}:={(\bigwedge {(M_{2n}^{+})}^{⁎}\otimes M_{2n}^{+})}^G$ doesn't satisfy a similar property.