Lie Groups Seminar
The first manifestly positive rule for multiplying Schubert classes in the
cohomology of Grassmannians, due to Littlewood and Richardson, dates
from the 1930s (but was only proven in the '70s). I conjectured in 1999
an extension to d-step flag manifolds (Grassmannians being d=1),
which was too naive for d=3, but proven for d=2 by [BKPT] in 2014,
and conjecturally corrected for d=3 by Buch in 2006.
I'll reinterpret the puzzle rule in equivariant cohomology in terms of
contracting tensors, one of which looks suspiciously like an R-matrix
(I'll explain what that means). The proven and conjectural puzzle rules
suggest which R-matrices to dig out of the literature, and with
degenerate versions we confirm the conjecture for H(3-step flag manifolds).
In addition, we prove rules for K(2-step and 3-step), K_T(2-step),
for neither of which was there even an extant conjecture.
This work is joint with Paul Zinn-Justin. In a future talk I'll
explain how the nondegenerate R-matrices may be interpreted as
Schubert calculus on certain quiver varieties.