Seminar on Combinatorics, Lie Theory, ​and Topology

Charge, Atoms and Crystals in Representation Theory

by Leonardo Patimo (Albert-Ludwigs-Universität Freiburg)

Aula Magna (Department of Mathematics)

Aula Magna

Department of Mathematics


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.