arXiv subjects

Salvatore Dolce

Publications and source records attributed to Salvatore Dolce.

2 recordsLinked to original sources

arXiv2015

On covariants in exterior algebras for the even special orthogonal group

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.

math.RA↗

arXiv2015

On certain modules of covariants in exterior algebras

We study the structure of the space of covariants $B:=\left(\bigwedge (\mathfrak g/\mathfrak k)^*\otimes \mathfrak g\right)^{\mathfrak k},$ for a certain class of infinitesimal symmetric spaces $(\mathfrak g,\mathfrak k)$ such that the space of invariants $A:=\left(\bigwedge (\mathfrak g/\mathfrak k)^*\right)^{\mathfrak k}$ 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 $\left(\bigwedge (M_n^{\pm})^*\otimes M_n\right)^G$, 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.

math.RA↗