A New Combinatorial Model in Representation Theory

The present paper is a survey of a simple combinatorial model for the irreducible characters of complex semisimple Lie algebras, and, more generally, of complex symmetrizable Kac-Moody algebras. This model, which will be referred to as the alcove path model, can be viewed as a discrete counterpart to the Littelmann path model. It allows us to give character formulas and a LittlewoodRichardson rule for decomposing tensor products of irreducible representations; it also leads to a nice description of crystal graphs, including a combinatorial realization of them as self-dual posets via a generalization of Schützenberger’s involution. Overall, we can say that the alcove path model leads to an extensive generalization of the combinatorics of irreducible characters from Lie type A (where the combinatorics is based on Young tableaux, for instance) to arbitrary type.