Partial and global representations of finite groups

TL;DR

This paper develops a systematic theory of partial representations of finite groups, focusing on those restricting to global representations of a subgroup, with explicit computations and applications to symmetric groups.

Contribution

It extends classical representation theory by introducing a framework for partial representations and providing explicit computational tools for finite groups and their subgroups.

Findings

Extended classical representation theory to partial representations.

Developed explicit computational methods for partial representations.

Applied theory to symmetric groups and specific subgroups.

Abstract

Given a subgroup H of a finite group G, we begin a systematic study of the partial representations of G that restrict to global representations of H. After adapting several results from [DEP00] (which correspond to the case where H is trivial), we develop further an effective theory that allows explicit computations. As a case study, we apply our theory to the symmetric group and its subgroup of permutations fixing 1: this provides a natural extension of the classical representation theory of the symmetric group.