The second fundamental theorem of invariant theory for the orthosymplectic supergroup

TL;DR

This paper proves the second fundamental theorem of invariant theory for the orthosymplectic supergroup using algebraic supergeometry, extending classical results and providing new proofs independent of Capelli identities.

Contribution

It establishes the second fundamental theorem for the orthosymplectic supergroup via algebraic supergeometry, linking it to Brauer diagrams and general linear supergroup cases.

Findings

Proves the second fundamental theorem for the orthosymplectic supergroup.

Provides new proofs for classical orthogonal and symplectic groups.

Replaces Capelli identities with algebraic geometric arguments.

Abstract

In a previous work we established a super Schur-Weyl-Brauer duality between the orthosymplectic supergroup of superdimension $(m|2n)$ and the Brauer algebra with parameter $m-2n$. This led to a proof of the first fundamental theorem of invariant theory, using some elementary algebraic supergeometry, and based upon an idea of Atiyah. In this work we use the same circle of ideas to prove the second fundamental theorem for the orthosymplectic supergroup. The proof uses algebraic supergeometry to reduce the problem to the case of the general linear supergroup, which is understood. The main result has a succinct formulation in terms of Brauer diagrams. Our proof includes new proofs of the corresponding second fundamental theorems for the classical orthogonal and symplectic groups, as well as their quantum analogues. These new proofs are independent of the Capelli identities, which are replaced by algebraic geometric arguments.