The first fundamental theorem of invariant theory for the orthosymplectic supergroup

Abstract

We give an elementary proof of the first fundamental theorem of the invariant theory for the orthosymplectic supergroup by generalising the method of Atiyah, Bott and Patodi to the supergroup context. We use methods from super-algebraic geometry to convert invariants of the orthosymplectic supergroup into invariants of the corresponding general linear supergroup on a different space. In this way, a super Schur-Weyl-Brauer duality is established between the orthosymplectic supergroup of superdimension $(m|2n)$ and the Brauer algebra with parameter $m-2n$. The result may be interpreted in terms of the relevant Harish-Chandra super pair action (over the complex field), or equivalently, the orthosymplectic Lie supergroup over the infinite dimensional Grassmann algebra. We also state a corresponding theorem for the orthosymplectic Lie superalgebra, which involves an extra invariant generator, the super-Pfaffian.