Rank reduction of string C-group representations

TL;DR

This paper generalizes a rank reduction technique for string C-group representations, enabling the construction of high-rank representations for various groups and suggesting the uniqueness of certain groups with rank gaps.

Contribution

It extends the rank reduction method to arbitrary settings and applies it to orthogonal groups, broadening the scope of constructing high-rank string C-group representations.

Findings

Orthogonal groups over fields of even order > 2 have string C-group representations of all ranks 3 to d.

The rank reduction technique is broadly applicable to different group families.

The group ${ m Alt}(11)$ may be more unique in having rank gaps than previously believed.

Abstract

We show that a rank reduction technique for string C-group representations first used for the symmetric groups generalizes to arbitrary settings. The technique permits us, among other things, to prove that orthogonal groups defined on $d$-dimensional modules over fields of even order greater than 2 possess string C-group representations of all ranks $3\leq r\leq d$. The broad applicability of the rank reduction technique provides fresh impetus to construct, for suitable families of groups, string C-groups of highest possible rank. It also suggests that the alternating group ${\rm Alt}(11)$---the only known group having `rank gaps'---is perhaps more unusual than previously thought.