Department of Mathematics, Aula Magna.
The dimensions of the weight spaces of irreducible representations of reductive groups can be q-deformed, obtaining the Kostka-Foulkes polynomials, which measure the dimensions of the Brylinski-Kostant filtration and therefore have positive coefficients. Lascoux and Schützenberger gave in ’78 a combinatorial interpretation of the coefficients of these polynomials in type A, known as charge. However, finding such an interpretation for other groups is still an open problem. In a new approach, based on the geometry of the affine Grassmannian, we constructed the charge in terms of the associated crystal graph starting from a decomposition of the crystal into atoms. This allowed us to recover geometrically the results of Lascoux and Schützenberger in type A and additionally, with a similar approach we constructed a charge in type C2 (joint with J. Torres), obtaining for the first time a charge outside of type A.
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