Algebra Seminar: Zhang -- Second fundamental theorem of invariant theory

Yang Zhang (University of Sydney) 

Friday 20 May, 12-1pm, Place: Carslaw 375 

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)-> End_OSp(V)(V^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.