Alcove walks, buildings, symmetric functions and representations
For a complex simple Lie algebra g, the dimension K�µ of theweight space of a finite dimensional representation of highest weight λ is the same as the number of Littelmann paths of type λ and weight � . In this paper we give an explicit construction of a path of type λ and weightwhenever K�µ 6 0. This construction has additional consequences, it produces an explicit point in the building which chamber retracts to λ and sector retracts to � , and an explicit point of the affine Grassmannian in the corresponding Mirkovic-Vilonen intersection. In an appendix we discuss the connection between retractions in buildings and alcove walks.