Chapter 2. Projective representations and spin characters of complex reflection groups $G(m,p,n)$ and $G(m,p,\infty)$, I

Abstract

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.