Update: Notes and video from the summer school is available here.
I should have been posting while I was at the conference. But anyway I’ll try to post as much as I can remember, before I forget it.
Geometric Satake gives a correspondence from representation theory to subvarieties of an affine Grassmanian. MV-cycles help give a better handle on them but are still rather abstract. MV-polytopes , introduced by Jared Anderson, are more “hands on” in terms of being able to do direct computations. But you need to know what they are first, and their original definition as moment map images of MV-cycles doesn’t really help. At least if you know they are a convex hill of the torus fixed-points appear in the *closure* of an MV-cycle then you’re done — but this still requires more-or-less calculating the MV-cycles and taking their closure.
Kamnitzer’s thesis provided a few ways to get direct handle on MV-polytopes avoiding MV-cycles entirely.
- Implicit description via Plucker relations
- Inductive description (for
later extended to types B and C by **FIXME**)
- Reduction to dim-2: higher dimensional MV-polytopes are all polytopes whose 2-faces are MV-polytopes.
- Construction from primitives: MV-polytopes are Minkowski sums of “primitives” and sums of primitives from the same “cluster” are MV-polytopes
- In dimension 2, clusters can be described by networks of non-overlapping cords each parallel to a sides of the Weyl polytope
