Combinatorial Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 17 September 2013

Appendix B

B2. Complex reflection groups

A complex reflection is an invertible linear transformation of ${ℂ}^n$ of finite order which has exactly one eigenvalue that is not 1. A complex reflection group is a group generated by complex reflections in ${ℂ}^n\text{.}$ The finite complex reflection groups have been classified by Shepard and Todd [STo1954]. Each finite complex reflection group is either

(a)	$G(r,p,n)$ for some positive integers $r,$ $p,$ $n$ such that $p$ divides $r,$ or
(b)	one of 34 other “exceptional” finite complex reflection groups.

Let $r,p,d$ and $n$ be positive integers such that $pd=r\text{.}$ The complex reflection group $G(r,p,n)$ is the set of $n\times n$ matrices such that

(a)	The entries are either 0 or $r\text{th}$ roots of unity,
(b)	There is exactly on nonzero entry in each row and each column,
(c)	The $d\text{th}$ power of the product for the nonzero entries is 1.

The group $G(r,p,n)$ is a normal subgroup of $G(r,1,n)$ of index $p$ and $$\left|G(r,p,n)\right|=d{r}^{n-1}n!\text{.}$$ In addition

(a)	$G(1,1,n)\cong S_n$ the symmetric group or Weyl group of type $A_{n-1}$,
(b)	$G(2,1,n)$ is the hyperoctahedral group or Weyl group of type $B_n,$
(c)	$G(r,1,n)\cong (\mathbb{Z}/r\mathbb{Z})\wr S_n,$ the wreath product of the cyclic group of order $r$ with $S_n,$
(d)	$G(2,2,n)$ is the Weyl group of type $D_n\text{.}$

Partial results for $G(r,1,n)$

The following are answers to the main questions (Ia-c) for the groups $G(r,1,n)\cong (\mathbb{Z}/r\mathbb{Z})\wr S_n\text{.}$ For the general $G(r,p,n)$ case see [HRa1998].

I. What are the irreducible $G(r,1,n)\text{-modules?}$

(a)	How do we index/count them?
	There is a bijection $$\genfrac{}{}{0ex}{}{r\text{-tuples}\hspace{0.17em}\lambda =({\lambda }^{\left(1\right)},\dots ,{\lambda }^{\left(r\right)})\hspace{0.17em}\text{of partitions}}{\text{such that}\hspace{0.17em}{\sum }_{i=1}^r\left|{\lambda }^{\left(i\right)}\right|=n}\stackrel{1-1}{⟷}\text{Irreducible representations}\hspace{0.17em}{C}^{\lambda }\text{.}$$
(b)	What are their dimensions?
	The dimension of the irreducible representation $C^{\lambda }$ is given by $$\begin{array}{ccc}\text{dim}\left(C^{\lambda }\right)& =& \text{\# of standard tableaux of shape}\hspace{0.17em}\lambda \\ & =& n!\prod _{i=1}^r\prod _{x\in {\lambda }^{\left(i\right)}}\frac{1}{h_x},\end{array}$$ where $h_x$ is the hook length at the box $x\text{.}$ A standard tableau of shape $\lambda =({\lambda }^{\left(1\right)},\dots ,{\lambda }^{\left(r\right)})$ is any filling of the boxes of the ${\lambda }^{\left(i\right)}$ with the numbers $1,2,\dots ,n$ such that the rows and the columns of each ${\lambda }^{\left(i\right)}$ are increasing.
(c)	What are their characters?
	A Murnaghan-Nakayama type rule for the characters of the groups $G(r,1,n)$ was originally given by Specht [Spe1932]. See also [Osi1954] and [HRa1998].

References

The original paper of Shepard and Todd [STo1954] remains a basic reference. Further information about these groups can be found in [HRa1998]. The articles [OSo1980], [Leh1995], [Ste1989-2], [Mal1995] contain other recent work on the combinatorics of these groups.

Notes and references

This is the survey paper Combinatorial Representation Theory, written by Hélène Barcelo and Arun Ram.

Key words and phrases. Algebraic combinatorics, representations.

Barcelo was supported in part by National Science Foundation grant DMS-9510655.
Ram was supported in part by National Science Foundation grant DMS-9622985.
This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..

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