## University of Sydney Algebra Seminar

# Anthony Henderson (University of Sydney)

## Friday 5 October, 12-1pm, Place: Carslaw 375

## Complementary symmetry for Slodowy varieties of classical groups

Let λ and μ be two partitions with the first dominating the second.
Various quantities f(λ,μ) attached to such a pair are significant in representation theory,
and often they obey a rule of "complementary symmetry": if λ^{c} and μ^{c} denote the
partitions obtained by taking the diagrams complementary to those of λ and μ in
a fixed rectangle and rotating them 180 degrees, then f(λ^{c},μ^{c})=f(λ,μ).
For example, this holds for the Kostka number K_{λ,μ} describing weight multiplicities of
irreducible representations of the general linear group, and for the decomposition numbers
d_{λ,μ} describing the reduction of those irreducible representations modulo a prime.
I will recall a geometric result which I published in 2015, showing that Slodowy varieties for the
general linear Lie algebra obey this complementary symmetry (which implies both of the aforementioned examples),
and explain a recent generalization to other classical groups due to Yiqiang Li.