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.