String $C$-groups with real Schur index $2$

TL;DR

This paper presents examples of finite string C-groups, which are automorphism groups of abstract regular polytopes, exhibiting irreducible characters with a real Schur index of 2, thus addressing a specific open problem.

Contribution

It provides the first known examples of such groups with irreducible characters of real Schur index 2, solving a problem posed by Monson.

Findings

Existence of finite string C-groups with real Schur index 2

Examples of automorphism groups of regular polytopes with specific character properties

Addresses an open problem in the theory of string C-groups

Abstract

We give examples of finite string $C$-groups (the automorphism groups of abstract regular polytopes) that have irreducible characters of real Schur index $2$. This answers a problem of Monson concerning these groups.