Supercharacter theories for discrete algebra groups

Abstract

The goal of this thesis is the extension of a construction of a supercharacter theory (first established for finite groups) to the context of infinite countable discrete groups, namely, for amenable countable discrete algebra groups. By an algebra group we mean a group of the form G = 1+A where A is an associative nil algebra over a field K, which generalize the group Un(K) consisting of all unipotent uppertriangular n_n-matrices over K. We may think about a supercharacter theory for a finite group as an approximation of the usual irreducible character theory, and it as been proved to provide a rich alternative to deal with the group representation theory. The success of supercharacter theories for finite groups motivates its generalization to infinite countable discrete groups, since there is a well defined character theory for these groups. We develop a standard supercharacter theory that simultaneously extends the standard finite supercharacter theory, and allows us to deal with different types of algebra groups (depending on the the characteristic of K and on the K-dimension of A) for which “typical” approaches do not work. Our supercharacter theory translates into an ergodic framework, where supercharacters are defined by certain ergodic measures on the Pontryagin dual group of the abelian additive group A+. This identification makes possible to present, not only integral expressions (over orbital closures) for supercharacter values, but also canonical unitary representations affording supercharacters. We pay special attention to algebra groups realized as direct limits of finite algebra groups, which are locally nilpotent groups. For these groups, there is an innermost relationship with the finite standard supercharacter theory. Furthermore, our supercharacter theory establishes a link between the usual methods used when dealing either with nilpotent discrete groups or direct limits of finite groups. This is exemplified with the two infinite unitriangular groups of positive characteristic: the unitriangular group Un(F) over an algebraic closed field of prime characteristic, and the locally finite unitriangular group U∞(Fq) over a finite field.