Representation Degrees, Character Values and Applications

In recent years there has been a growing use of Representation Theory in Asymptotic Group Theory and related topics.On the one hand, there is the rather new subject of representation growth, somewhat analogous to the classical study ofsubgroup growth. On the other hand, the solutions of various seemingly unrelated problems - such as random walks on groups,or random generation of simple groups, or word maps on groups - require new results in representation theory, involving character degreesand values. In this proposal we plan to study several quantitative aspects of representation theory, and to applythem in a wide variety of such contexts.

The first part of the proposal is concerned with the study of representation growth of various classes of groups via their representation zeta functions. This is analogous to, and in a sense a generalization of, the well developed theory of subgroup growth. Subgroup growth is a rich subject with many ramifications and applications not just in group theory, but also in related areas such as number theory. Representation growth, although not nearly as well developed as subgroup growth, also has ramifications in number theory, and in addition has strong connections to geometry. Indeed, the first occurrence of a representation zeta-function was in a paper of Witten, where he proved that for a compact Lie group the value of the representation zeta function at an even integer is equal to the volume of a natural moduli space associated to the group. Since then, representation zeta functions have arisen naturally in a number of similar geometrical contexts, for example for finite groups of Lie type such as GL(n,q). Advances in our knowledge of representation growth could have an impact on these areas of geometry. The second part of the proposal is concerned with character values of finite groups of Lie type, and applications. One particular application is to the theory of random walks on groups, a branch where the fields of group theory and probability meet. The main conjecture in the proposal would, if proved, have wonderful consequences for the theory of random walks, and consequently a substantial impact on this branch of probability.