# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1812.01055