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 $k=[n/2]$.

Let $G(m,p,n)$ be a complex reflection group and $R\big(G(m,p,n)\big)$ one of its representation group, and let $G(m,p,\infty)$ and $R\big(G(m,p,\infty)\big)$ be their inductive limits as $n\to \infty$. We study projective irreducible representations (= IRs) of $G(m,p,n)$ and their characters which we call spin characters of them. We study in particular projective IRs of generalized symmetric groups $G(m,1,n)$ and projective factor representations of $G(m,1,\infty)$ and their characters, and also limiting process as $n\to\infty$. Since $R\big(G(m,1,n)\big)$ is a special central extension of $G(m,1,n)$ by the Schur multiplier $Z=H^2\big(G(m,1,n), \boldsymbol{C}^\times\big)$, a projective IR $\pi$ of $G(m,1,n)$ has its spin type, a character $\chi$ of $Z$, such that $\pi(z) = \chi(z)I$. In the latter part of the paper we study in detail the case of a certain spin type and also the relation to the non-spin case.

In this paper we study projective (or spin) irreducible representations and their characters of generalized symmetric groups $G(m,1,n)$, and spin characters of their inductive limit groups $G(m,1,\infty)=\lim_{n\to\infty}G(m,1,n)$. The groups $G(m,1,n)$ form a subcategory of complex reflection groups $G(m,p,n),\,p|m$, and the present study has a fundamental importance for such studies for general $G(m,p,n)$'s. Schur multipliers $Z=H^2\big(G(m,1,n),\boldsymbol{C}^\times\big)$ are isomorphic to $\boldsymbol{Z}_2^{\;3}=\prod_{1\le i\le 3}\langle z_i\rangle,\, z_i^{\,2}=e,$ for $n\ge 4$ and $m\ge 2$ even, and similarly for $n=\infty$. Here, according to the semidirect product structure $G(m,1,n)=D_n\rtimes\mathfrak{S}_n$ with $D_n=\boldsymbol{Z}_m^{\;n}$, $z_1$ corresponds to the double covering group $\widetilde{\mathfrak{S}}_n$ of $\mathfrak{S}_n$, and $z_2$ to the double covering $\widetilde{D}_n$ of $D_n$, and $z_3$ to the twisted action of $\widetilde{\mathfrak{S}}_n$ on $\widetilde{D}_n$. In this case, any such representations and such characters have their own central characters $\chi\in\widehat{Z}$ with $(\beta_1,\beta_2,\beta_3),\,\beta_i=\chi(z_i)=\pm 1$, called (spin) type. Our study here is for two types $(-1,-1,-1)$ and $(-1,-1,\;1)$, and gives (1) classification and construction of all spin irreducible representations of $G(m,1,n)$, (2) calculation of their characters, (3) calculation of limits of normalized irreducible characters as $n\to \infty$, and (4) explicit determination of all the spin characters of $G(m,1,\infty)$ of these types.