Random Walks in Weyl Chambers and the Decomposition of Tensor Powers

We consider a class of random walks on a lattice, introduced by Gessel and Zeilberger, for which the reflection principle can be used to count the number of k-step walks between two points which stay within a chamber of a Weyl group. We prove three independent results about such reflectable walks: first, a classification of all such walks; second, many determinant formulas for walk numbers and their generating functions; third, an equality between the walk numbers and the multiplicities of irreducibles in the kth tensor power of certain Lie group representations associated to the walk types. Our results apply to the defining representations of the classical groups, as well as some spin representations of the orthogonal groups.