Applications of polytopes to the representation theory of Lie algebrasTyrrell McAllister, Wyoming |
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November 12, 2009 AbstractPolyhedral geometry has long provided insight into the representations of Lie algebras. An early example of the polyhedral approach was the introduction of the Gelfand--Tsetlin polytopes in the 1950's. More recently, Berenstein and Zelevinsky in 2001 encoded tensor product multiplicities for semi-simple Lie algebras as the number of integer lattice points in certain families of polyhedra. These results have theoretical implications as well as concrete computational applications. We discuss a particular combinatorial structure on Gelfand--Tsetlin polytopes, which we call their tiling poset. We use this combinatorial structure to compute the degrees of so-called "stretched Kostka coefficients", which express the dimension of a weight space as the parametrizing weights are scaled by an integer parameter. We also discuss applications of polyhedral algorithms to the computational complexity of tensor-product multiplicities. We mention some conjectures motivated by the polyhedral interpretation of these coefficients, including a conjecture that, if true, generalizes the Saturation Theorem of A. Knutson and T. Tao. |