Séminaire SPACE Tours
# Charge, Atoms and Crystals in Representation Theory

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E2 1180 (Tours)
### E2 1180

#### Tours

Description

One of the most fundamental problems in representation theory is to

compute the dimensions of the weight spaces of the irreducible

representations of groups. In the case of a reductive group (for

example GL_n(C)) there are many combinatorial models that answer this

question, for example by counting the number of semistrandard

tableaux. The dimensions of the weight spaces of irreducible representations of

reductive groups have a natural q-deformation, called the

Kostka-Foulkes polynomials. However, finding combinatorial statistics

which express these polynomials is a long-standing open problem in

algebraic combinatorics, which has only been solved in type A.

We develop a new approach based on the geometry of the affine

Grassmannian, where we construct the charge in terms of the associated

crystal graph, starting from a decomposition of the crystal into

atoms. This not only allows us to recover the results of Lascoux and

Schützenberger in type A geometrically, but also to define a charge

statistic in type C2 (joint with J. Torres).