The University of Michigan Combinatorics Seminar
Fall 2003
September 12, 4:10-5:00, 3866 East Hall





Littelmann paths, minuscule representations, and affine Lie algebras

Peter Magyar

Michigan State University




Abstract

Littelmann's path model gives a flexible way to construct the crystal graph of any irreducible representation of a Kac-Moody algebra: almost all the known generalizations of Young tableaux can be derived from it.

We focus on the case of an affine Lie algebra g-hat and its basic representation V(Lam_0). When the underlying finite-dimensional algebra g is of classical type, the Kyoto school has given a model for the crystal graph in terms of certain infinite paths. We unify this picture with Littelmann's finite paths, and extend the Kyoto model to g = E_6 and E_7: namely to all those g which admit a minuscule representation.

Our model gives a combinatorial explanation of the "Tensor Product Phenomenon": The g-hat module V(Lam_0), when considered as a g-module, is isomorphic to an infinite tensor product of small representations of g.

Reference: math.RT/0308156


VIGRE Pre-talk

The above talk will be preceded by a VIGRE pre-talk, also by Peter Magyar, to be held from 3:15-3:45pm in 3866 East Hall, i.e. the regular seminar room. The topic of the pre-talk is:

The Littelmann Path Model