## A polytope calculus for semisimple groups.(English)Zbl 1064.20047

Summary: We define a collection of polytopes associated to a semisimple group $$\mathbf G$$. Weight multiplicities and tensor product multiplicities may be computed as the number of such polytopes fitting in a certain region. The polytopes are defined as moment map images of algebraic cycles discovered by I. Mirković and K. Vilonen. These cycles are a canonical basis for the intersection homology of (the closures of the strata of) the loop Grassmannian.

### MSC:

 20G05 Representation theory for linear algebraic groups 14L35 Classical groups (algebro-geometric aspects) 20G10 Cohomology theory for linear algebraic groups 52B12 Special polytopes (linear programming, centrally symmetric, etc.)
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### References:

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