| 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 classify the cases in which these techniques can be used, and apply them to obtain determinant formulas in many cases. We also prove an equality between these walk-numbers and the multiplicities of irreducibles in the kth tensor power of certain Lie group representations associated to the walk-types; this result applies for the defining representations of all the classical groups. We also generalize the argument to the continuous case, which gives determinant formulas for Brownian motion in Weyl chambers with either absorbing or reflecting boundary conditions.;Another application of walks to representation theory is the use of walks to construct representations. We define Hasse walk algebras on a poset P, algebras which act on linear combinations of walks from 0 to rank n in P. A series of such algebras has P as its Bratelli diagram. We study constructions of these algebras with generators which act locally, changing only one step of a walk in a way which depends only on the adjacent steps. We prove general existence and construction results, and then construct specific examples. In particular, for the Boolean algebra ;We also study the analogous problem in which the walks of length n are allowed to go either up or down. This gives a natural construction of the Temperly-Lieb algebra. The construction is also natural on the Young-Fibonacci lattice YF, a lattice defined by Stanley which has many of the same combinatorial properties as Young's lattice; the resulting algebra is analogous to the Brauer algebra on Young's lattice. We also construct a Robinson-Schensted algorithm which is specific to YF and which has interesting combinatorial properties.;In one case, the determinants can be factored; this leads to an equality between two related walk-numbers, which also has an interpretation in representation theory. We then use the equality for walks to obtain a combinatorial correspondence. |
