University of Sydney Algebra Seminar

Ruibin Zhang (University of Sydney)

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

Invariants of the orthosymplectic Lie superalgebra and the super Pfaffian

The invariant theory of the orthosymplectic Lie superalgebra $\mathfrak{osp}(V)$ is more intricate than that of the orthosymplectic supergroup $OSp(V)$. For example, the endomorphism algebra $En{d}_{OSp(V)}(V^{\otimes r})$ over the orthosymplectic supergroup is a quotient of the Brauer algebra of degree $r$ (with parameter equal to the superdimension of $V$) as proved recently, but the endomorphism algebra $En{d}_{\mathfrak{osp}(V)}(V^{\otimes r})$ over the orthosymplectic superalgebra contains elements which are not Brauer-like. We will describe explicitly some $\mathfrak{osp}(V)$-invariants which are referred to as super Pfaffians, and show that the $OSp(V)$-invariants and super Pfaffians together generate all the $\mathfrak{osp}(V)$-invariants. This is joint work with Gus Lehrer.