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Title:聽Perfect bases in representation theory.

Abstract: A famous foundational problem concerns finding combinatorial expressions for tensor products of irreducible representations. A conceptually satisfying way to answer this question is to find bases for representations which restrict to bases of tensor product multiplicity spaces. These bases are called "good" or "perfect" and were first proposed 35 years ago by Gelfand and Zelevinsky. The construction of such bases is more difficult than one might expect and cannot be achieved by elementary means: they require geometric inputs such as the geometric Satake correspondence (Mirkovic-Vilonen), or the theory of perverse sheaves on quiver varieties (Lusztig). These lead to the MV basis, and Lusztig's dual canonical and dual semicanonical bases.
Remarkably, each such perfect basis gives rise to the same combinatorial structure, which is encoded as a collection of polytopes. With Pierre Baumann and Allen Knutson, we defined measures supported on these polytopes. These measures allow us to perform computations which distinguish among these different bases.