November 19, 2007, 2:10pm

Bahen 6183

Abstract:The classical construction of Gelfand and Cetlin associates a polytope to each irreducible representation of GL(n, C) such that the integral points in the polytope parameterize a natural basis for the representation. Guillemin and Sternberg gave a symplectic geometric realization of G-C polytopes, namely, the Gelfand-Cetlin polytope is the image of an (almost everywhere differentiable) moment map on the flag variety. The G-C construction has been generalized to any (complex) reductive algberaic group via representation theoretic methods by Littlemann, Bernstein and Zelevinsky. These are called string polytopes. We will discuss these as well as a recent result of mine which realizes the string polytopes as the Newton polytopes for flag varieties. Unfortunately, there is no generalization of Guillemin-Sternberg for arbitrary reductive groups and string polytopes. (though there has been some nice partial results e.g. Megumi Harada's result for symplectic group). We will give a suggestion for a general construction using toric degenerations.