Ruibin Zhang: First and Second Fundamental Theorems of Invariant Theory for Classical Supergroups

We present the first and second fundamental theorems of invariant theory for the general linear and orthosymplectic supergroups. We will first discuss the invariant theory of the general linear supergroup, then generalise the method of Atiyah, Bott and Patodi to the supergroup context to prove a super Schur-Weyl-Brauer duality between the orthosymplectic supergroup of superdimension (m|2n) and the Brauer algebra with parameter m-2n. Using similar ideas, we prove the second fundamental theorem for the orthosymplectic supergroup by reducing the problem to the known case of the general linear supergroup. Our main results have a succinct formulation in terms of Brauer diagrams, and recover the fundamental theorems of invariant theory for the orthogonal and symplectic groups as special cases. This is joint work with Gus Lehrer.