Alcove walks, buildings, symmetric functions and representations

Abstract

For a complex simple Lie algebra, the dimension $K_{λμ}$ of the $μ$ weight space of a finite dimensional representation of highest weight $λ$ is the same as the number of Littelmann paths of type $λ$ and weight $μ$. In this paper we give an explicit construction of a path of type $λ$ and weight $μ$ whenever $K_{λμ}\ne 0$. This construction has additional consequences, it produces an explicit point in the building which chamber retracts to $λ$ and sector retracts to $μ$, and an explicit point of the affine Grassmannian in the corresponding Mirković-Vilonen intersection. In an appendix we discuss the connection between retractions in buildings and alcove walks.