Nonlinearity of matrix groups

TL;DR

This paper investigates the linearity properties of matrix groups over rings, specifically determining conditions under which the group $EL_3(R)$ admits faithful finite-dimensional complex representations.

Contribution

It establishes a criterion linking the existence of faithful finite-dimensional representations of $EL_3(R)$ to the properties of finite index ideals in the ring $R$, answering a question posed by Guoliang Yu.

Findings

$EL_3(R)$ has a faithful finite-dimensional complex representation iff $R$ has a finite index ideal with such a representation.

The result applies to any unitary associative ring, providing a broad criterion.

The paper clarifies the nonlinearity of certain matrix groups over infinite rings.

Abstract

The aim of this note is to answer a question by Guoliang Yu of whether the group $EL_3(Z<x,y>)$, where $Z<x,y>$ is the free (non-commutative) ring, has any faithful linear representations over a field. We prove, in particular, that for every (unitary associative) ring $R$, the group $EL_3(R)$ has a faithful finite dimensional complex representation if and only if $R$ has a finite index ideal that has a faithful finite dimensional complex representation.