Chapter 1. Introductory expositions on projective representations of groups

Abstract

This paper gives introductory expositions on the theory of projective (or spin) representations of groups, together with plotting historical milestones of the theory, starting from Schur's trilogy in 1904, 1907 and 1911, Cartan's work in 1913, Pauli's introduction of spin quantum number in 1925, and Dirac's relativistic equation of electron in 1928, and so on. We pick up many situations where multi-valued representation of groups appear naturally, and thus explain how the projective representations are indispensable and worth to study. We discuss rather in detail the case of Weil representations of the symplectic groups $Sp(2n,\boldsymbol{R})$, and the case of various actions of the symmetric group $\mathfrak{S}_n$ on the full matrix algebra $M(2^k,\boldsymbol{C})$ of degree $2^k$ with $k=[n/2]$.