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 λc and μc 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(λcc)=f(λ,μ). For example, this holds for the Kostka number Kλ,μ describing weight multiplicities of irreducible representations of the general linear group, and for the decomposition numbers dλ,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.