IDeAS Seminar: A random walk on the upper triangular matrices

We study the following random walk on the group of n n upper triangular matrices with coecients in Z=pZ and ones along the diagonal. Starting at the identity, at each step we choose a row at random and either add it to or subtract it from the row immediately above. The mixing time of this walk is conjectured to be asymptotically n2p2. While much has been proven in this direction by a number of authors, the full conjecture remains open. We sharpen the techniques introduced by Arias-Castro, Diaconis, and Stanley to show that the dependence on p of the mixing time is p2. To prove this result, we use super-character theory and comparison theory to bound the eigenvalues of this random walk.