Chapter 3. Projective representations and spin characters of complex reflection groups $G(m,p,n)$ and $G(m,p,\infty)$, II: Case of generalized symmetric groups

Abstract

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.