Maximal subgroups of free idempotent generated semigroups over the full linear monoid

TL;DR

This paper proves that certain maximal subgroups of free idempotent generated semigroups over the full linear monoid are isomorphic to general linear groups, specifically when the rank is less than one-third of the matrix size.

Contribution

It establishes an isomorphism between maximal subgroups of these semigroups and general linear groups under specific rank conditions.

Findings

Maximal subgroups are isomorphic to GL_r(Q)

The result holds for r < n/3

Provides new insights into the structure of free idempotent generated semigroups

Abstract

We show that the rank r component of the free idempotent generated semigroup of the biordered set of the full linear monoid of n x n matrices over a division ring Q has maximal subgroup isomorphic to the general linear group GL_r(Q), where n and r are positive integers with r < n/3.