We'll talk about how basic algebra (representation theory) can be used to determine fundamental questions about the rate of convergence of random walks.  We'll also discuss some of the difficulties in this approach, here focusing on joint work with  Daniel Bump, Persi Diaconis, Laurent Miclo, and Harold Widom studying a simple random walk on the Heisenberg group mod p (a particularly simple to describe noncommutative group).  Analysis of a random walk on the group dates back to Zach, who was considering the effectiveness of certain random number generators.  We'll assume some knowledge of linear algebra and a bit of basic group theory, but otherwise aim for an elementary talk.