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 $$End_{OSp(V)}(V^{\otimes r})$$ 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 $$End_{\mathfrak{osp}(V)}(V^{\otimes r})$$ 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.