| Abstract: We describe a new combinatorics for
representations of semisimple Lie groups, in terms of convex polytopes.
Although these polytopes arise from some geometry of Mirkovic and Vilonen
in the loop Grassmannian, we will focus on the combinatorics. We
describe an explicit conjectural construction of the polytopes that also
constructs the crystal graphs (where vertices are polytopes). Weight
multiplicities and tensor product multiplicities (Littlewood-Richardson
numbers) may be found by counting polytopes. We also discuss an algebra
spanned by the polytopes, where the product is related to Minkowski sum.
(This is joint work with Ivan Mirkovic.) |
