A rank augmentation theorem for rank 3 string C-group representations of the symmetric groups

TL;DR

This paper introduces a technique to increase the rank of certain symmetric group representations, specifically for rank 3 string C-groups, to rank 4, under specific conditions.

Contribution

It provides a new rank augmentation method for rank 3 string C-group representations of symmetric groups, expanding the understanding of their structure.

Findings

The technique successfully produces rank 4 representations from rank 3 under specified hypotheses.

Conditions for the validity of the augmented representations are clearly outlined.

The approach broadens the class of known string C-group representations for symmetric groups.

Abstract

We give a rank augmentation technique for rank 3 string C-group representations of the symmetric group $S_n$ and list the hypotheses under which it yields a valid string C-group representation of rank 4 thereof.