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. Sergeev has indicated how to define a “Pfaffian” in this space, and announced that, together with the invariants of the group $\mathrm{OSp}(V)$, it generates all invariants of ${\mathfrak {osp}}(V).$ We introduce a ‘space of super Pfaffians’, show it is a simple ${\mathfrak {gl}}_N$-submodule of $\mathcal {S}(N)^{{\mathfrak {osp}}(V)}$, give an explicit formula for its highest weight vector and prove that the super Pfaffians and the $\mathrm{OSp}(V)$-invariants 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 further analyse the module structure of that space. These results also enable us to determine the ${\mathfrak {osp}}(V)$-invariants in the tensor powers ${V}^{\otimes r}$ for all r.