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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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