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.