Representations of McLain groups

TL;DR

This paper studies basic modules of McLain groups, generalizing supercharacters, and provides their decomposition, irreducibility criteria, and conditions for multiplicity-free modules across various algebraic settings.

Contribution

It introduces a comprehensive framework for basic modules of McLain groups, including their construction, decomposition, and irreducibility criteria, extending previous supercharacter theories.

Findings

Basic modules are disjoint and their endomorphism algebras are characterized.

Conditions for basic modules to be irreducible are established.

Full decompositions of basic modules into irreducible components are provided.

Abstract

Basic modules of McLain groups $M=M(\Lambda,\leq, R)$ are defined and investigated. These are (possibly infinite dimensional) analogues of Andr\'e's supercharacters of $U_n(q)$. The ring $R$ need not be finite or commutative and the field underlying our representations is essentially arbitrary: we deal with all characteristics, prime or zero, on an equal basis. The set $\Lambda$, totally ordered by $\leq$, is allowed to be infinite. We show that distinct basic modules are disjoint, determine the dimension of the endomorphism algebra of a basic module, find when a basic module is irreducible, and exhibit a full decomposition of a basic module as direct sum of irreducible submodules, including their multiplicities. Several examples of this decomposition are presented, and a criterion for a basic module to be multiplicity-free is given. In general, not every irreducible module of a McLain group is a constituent of a basic module.