Projective (or spin) representations of finite groups. I

TL;DR

This paper introduces a practical method to construct representation groups of finite groups, facilitating the study of projective and spin representations, with applications to groups with specific Schur multipliers.

Contribution

It presents a new, efficient approach to explicitly construct representation groups for finite groups, aiding in the analysis of projective and spin representations.

Findings

Method successfully constructs representation groups for groups with prime Schur multiplier

Enables calculation of spin characters from linear representations of the constructed groups

Applied to several examples with Schur multiplier of order 3

Abstract

Schur multiplier $M(G)$ of a finite group $G$ has been studied heavily. To proceed further to the study of projective (or spin) representations of $G$ and their characters (called spin characters), it is necessary to construct explicitly a representation group $R(G)$ of $G$, a certain central extension of $G$ by $M(G)$, since projective representations of $G$ correspond bijectively to linear representations of $R(G)$. We propose here a practical method to construct $R(G)$ by repetition of one-step efficient central extensions according to a certain choice of a series of elements of $M(G)$. This method is also helpful for constructing linear representations of $R(G)$ and accordingly for calculating spin characters. Actually, we will apply this method to several examples of $G$ with prime number 3 in $M(G)$.