Projective (or spin) representations of finite groups. III

TL;DR

This paper develops a step-by-step method to explicitly construct and analyze the irreducible representations and spin characters of a specific finite group with a complex Schur multiplier, extending previous techniques to a more intricate case.

Contribution

It introduces a detailed, multi-step process for constructing representation groups and their irreducible representations for a finite group with a nontrivial Schur multiplier, using explicit methods and classical techniques.

Findings

Constructed the representation group of order 243 for G_{39}

Explicitly listed all irreducible representations of the group

Computed the spin characters of the representations

Abstract

In previous papers I and II under the same title, we proposed a practical method called Efficient stairway up to the Sky, and apply it to some typical finite groups $G$, with Schur multiplier $M(G)$ containing prime number 3, to construct explicitly their representation groups $R(G)$, and then, to construct a complete set of representatives of linear IRs of $R(G)$, which gives naturally, through sectional restrictions, a complete set of representatives of spin IRs of $G$. In the present paper, we are concerned mainly with group $G=G_{39}$ of order 27 in a list of Tahara's paper, with $M(G)={\mathbb Z}_3\times {\mathbb Z}_3$. In this case, to arrive up to the Sky, we have two steps of one-step efficient central extensions. By the 1st step, we obtain a covering group of order 81, and by the 2nd step we arrive to $R(G)$ of order 243. At the 1st step, to construct explicitly a complete set of representatives of IRs of the group, we apply Mackey's induced representations, and at the 2nd step, with a help of this result, we apply so-called classical method for semidirect product groups given by Hirai and arrive to a complete list of IRs of $R(G)$. Then, using explicit realization of these IRs, we can compute their characters (called spin characters).