Type: Seminar

Distribution: World

Expiry: 27 Jun 2016

CalTitle1: On the second fundamental theorem of invariant theory for the orthosymplectic supergroup

Auth: kevinc@pkevinc.pc (assumed)

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.