Mathematical Society of Japan Memoirs

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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Chapter information

Source
Takeshi Hirai, Akihito Hora, Etsuko Hirai, Projective Representations and Spin Characters of Complex Reflection Groups $G(m,p,n)$ and $G(m,p,\infty)$ (Tokyo: The Mathematical Society of Japan, 2013), 123-272

Dates
First available in Project Euclid: 13 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.msjm/1413220877

Digital Object Identifier
doi:10.2969/msjmemoirs/02901C030

Rights
Copyright © 2013, The Mathematical Society of Japan

Citation

Hirai, Takeshi; Hora, Akihito; Hirai, Etsuko. 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. Projective Representations and Spin Characters of Complex Reflection Groups $G(m,p,n)$ and $G(m,p,\infty)$, 123--272, The Mathematical Society of Japan, Tokyo, Japan, 2013. doi:10.2969/msjmemoirs/02901C030. https://projecteuclid.org/euclid.msjm/1413220877


Export citation