Alcove walks, buildings, symmetric functions and representations

Details Alcove walks, buildings, symmetric functions and representations Alcove walks, buildings, symmetric functions and representations

2008-07-23

arxiv.org/abs/0807.3602v1

Mathematics

Representation Theory

Scientific paper

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

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