## University of Sydney Algebra Seminar

# Jie Du (University of New South Wales)

## Friday 8th March, 12:05-12:55pm, Carslaw 373

### Representations of \(q\)-Schur superalgebras at a root of unity

I will report on a classification of irreducible representations over the \(q\)-Schur superalgebra at a root of unity. We simply apply the relative norm map introduced by P. Hoefsmit and L. Scott in 1977. This map is the \(q\)-analogue of the usual trace map which has many important properties related to Mackey decomposition, Frobenius reciprocity, Nakayama relation, Higman's criterion, and so on. By describing a basis for the \(q\)-Schur superalgebra in terms of relative norms, we may filter the algebra with a linear sequence of ideals associated with \(l\)-parabolic subgroups. In this way, we may attach a defect group to a primitive idempotent. Primitive idempotents with the trivial defect group can be classified by \(l\)-regular partitions, and others can be classified via Brauer homomorphisms.

This is joint work with H. Gu and J. Wang.