# Yang Zhang (University of Sydney)

## On the second fundamental theorem of invariant theory for the orthosymplectic supergroup

We study the second fundamental theorem (SFT) of invariant theory for the orthosymplectic supergroup OSp(V) within the framework of the Brauer category. Three main results are established concerning the surjective algebra homomorphism
$$F_r^r: B_r(m-2n)\rightarrow {\rm End}_{{\rm OSp(V)}}(V^{\otimes r})$$
from the Brauer algebra of degree $$r$$ with parameter $$m-2n$$ (the superdimension of V is $$(m|2n)$$) to the endomorphism algebra over OSp(V):
(1) We show that the minimal degree $$r$$ for which Ker $$F_r^r$$ is nonzero is equal to $$r_c=(m+1)(n+1)$$;
(2) The generators for Ker $$F_{r_c}^{r_c}$$ are constructed;
(3) The generators of Ker $$F_{r_c}^{r_c}$$ generate $$F_r^r$$ for all $$r>r_c.$$
In the special case $$m=1$$, we show that the kernel is generated by a single element $$E$$, and obtain an explicit formula for the generator. As an application, we provide uniform proofs for the main theorems in recent papers of Lehrer and Zhang on SFTs for the orthogonal and symplectic groups.