Projective representations and spin characters of complex reflection groups $G(m, p, n)$ and $G(m, p, \infty)$, III

TL;DR

This paper extends the classification and construction of spin representations and characters from finite complex reflection groups to their infinite limits, building on hereditary properties from previous work.

Contribution

It provides a detailed classification of irreducible projective representations for complex reflection groups and their infinite limits, expanding understanding of spin characters.

Findings

Classified irreducible projective representations of $G(m,1,n)$ for finite n.

Constructed spin characters of the inductive limit groups $G(m,1, )$.

Extended results to $G(m,p, )$ with $p|m$, $p>1$ for infinite groups.

Abstract

This paper is a continuation of two previous papers in MSJ Memoirs, Vol.\,29 (Math. Soc. Japan, 2013) with the same title and numbered as I and II. Based on the hereditary property given there, from mother groups $G(m,1,n)$, the generalized symmetric groups, to child groups $G(m,p,n)$, the complex reflection groups, we study in detail classification and construction of irreducible projective representations (= spin representations) and their characters of $G(m,1,n)$ for $n$ finite. Then, taking limits as $n$ tends to infinity, we obtain spin characters of the inductive limit groups $G(m,1,\infty)$. By the heredity studied further, this gives the main kernel of the results for $G(m,p,\infty)$ with $p|m, p>1$.