Abstract:
 The first and second fundamental theorems (FFT and SFT) of classical invariant theory are respectively concerned with generators and relations for invariants of group actions. Let G be the orthogonal group O(V) or the symplectic Sp(V), and let $\text{End}(V^{\otimes r})$ be the algebra of endomorphisms of $V \otimes ^{r}$ . The FFT of the invariant theory of G in this setting states that there is a surjective algebra homomorphism from the Brauer algebra of degree r to the subalgebra of invariants in End($V \otimes ^{r}$). However, the SFT remained elusive in this setting. We will develop an SFT by studying a category of Brauer tangle diagrams, and discuss the generalization of the results to the corresponding quantum groups. This is joint work with Gus Lehrer.
