An overview of the history of projective representations (spin representations) of groups

TL;DR

This paper provides a historical overview of the development of projective (spin) representations of groups, tracing contributions from early quaternion theory to modern mathematical and physical applications.

Contribution

It offers a comprehensive historical synthesis of the evolution of projective representations, connecting mathematical foundations with physical theories.

Findings

Historical connections between quaternion theory and projective representations

Key contributions from mathematicians like Schur, and physicists like Pauli and Dirac

Contextual understanding of the development of spin representations

Abstract

An overview of the history of projective representations (= spin representations) of groups, preceded by the prehistory of studies on the theory of quaternion due to Rodrigues and Hamilton. Beginning with Schur, we cover many mathematicians until today, and also physicists Pauli and Dirac. This is a self translation of Appendix A of my book "Introduction to the theory of projective representations of groups" in Japanese, 2018, Sugakushobo, and may serve as an introduction to our paper arXiv: 1804.06063 [math.RT] which will appear in Kyoto J. Math.