SMS scnews item created by Ulrich Thiel at Fri 28 Sep 2018 1207
Type: Seminar
Distribution: World
Expiry: 9 Nov 2018
Calendar1: 5 Oct 2018 1200-1300
CalLoc1: Carslaw 375
CalTitle1: Henderson -- Complementary symmetry for Slodowy varieties of classical groups
Auth: thiel@p526m.pc (assumed)
Algebra Seminar
Complementary symmetry for Slodowy varieties of classical groups
Henderson
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(λc,μc)=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.