Invariants of the orthosymplectic Lie superalgebra and super Pfaffians

Abstract

Given a complex orthosymplectic superspace $V$, the orthosymplectic Lie superalgebra $\mathfrak{osp}(V)$ and general linear algebra ${\mathfrak{gl}}_N$ both act naturally on the coordinate super-ring $\mathcal{S}(N)$ of the dual space of $V\otimes {\mathbb{C}}^N$, and their actions commute. Hence the subalgebra $\mathcal{S}(N{)}^{\mathfrak{osp}(V)}$ of $\mathfrak{osp}(V)$-invariants in $\mathcal{S}(N)$ has a ${\mathfrak{gl}}_N$-module structure. We introduce the space of super Pfaffians as a simple ${\mathfrak{gl}}_N$-submodule of $\mathcal{S}(N{)}^{\mathfrak{osp}(V)}$, give an explicit formula for its highest weight vector, and show that the super Pfaffians and the elementary (or `Brauer') $\mathrm{OSp}$-invariants together generate $\mathcal{S}(N{)}^{\mathfrak{osp}(V)}$ as an algebra. The decomposition of $\mathcal{S}(N{)}^{\mathfrak{osp}(V)}$ as a direct sum of simple ${\mathfrak{gl}}_N$-submodules is obtained and shown to be multiplicity free. Using Howe's $(\mathfrak{gl}(V),{\mathfrak{gl}}_N)$-duality on $\mathcal{S}(N)$, we deduce from the decomposition that the subspace of $\mathfrak{osp}(V)$-invariants in any simple $\mathfrak{gl}(V)$-tensor module is either $0$ or $1$-dimensional. These results also enable us to determine the $\mathfrak{osp}(V)$-invariants in the tensor powers $V^{\otimes r}$ for all $r$.

References in corpus (2)

• The second fundamental theorem of invariant theory for the orthogonal group
• The Brauer Category and Invariant Theory

Cited by in corpus (1)

• Invariants of the special orthogonal group and an enhanced Brauer category