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

B1. Coxeter groups, groups generated by reflections, and Weyl groups

A Coxeter group is a group $W$ presented by generators $S=\{s_1,\dots ,s_n\}$ and relations $$\begin{array}{cc}{s}_i^2=1,& \text{for}\hspace{0.17em}1\le i\le n,\\ {\left(s_i{s}_j\right)}^{m_{ij}}=1,& \text{for}\hspace{0.17em}1\le i\ne j\le n,\end{array}$$ where each $m_{ij}$ is either $\infty$ or a positive integer greater than 1.

A reflection is a linear transformation of ${ℝ}^n$ which is a reflection in some hyperplane.

A finite group generated by reflections is a finite subgroup of $GL(n,ℝ)$ which is generated by reflections.

Theorem B1.1. The finite Coxeter groups are exactly the finite groups generated by reflections.

A finite Coxeter group is irreducible if it cannot be written as a direct product of finite Coxeter groups.

Theorem B1.2. (Classification of finite Coxeter groups)

(a)	Every finite Coxeter group can be written as a direct product of irreducible finite Coxeter groups.
(b)	There is one irreducible finite Coxeter group corresponding to each of the following “types” $$A_{n-1},B_n,D_n,E_6,E_7,E_8,F_4,H_3,H_4,I_2\left(m\right)\text{.}$$

The irreducible finite Coxeter groups of classical type are the ones of types $A_{n-1},$ $B_n,$ and $D_n$ and the others are the irreducible finite Coxeter groups of exceptional type.

(a)	The group of type $A_{n-1}$ is the symmetric group $S_n\text{.}$
(b)	The group of type $B_n$ is the hyperoctahedral group $(\mathbb{Z}/2\mathbb{Z})\wr S_n,$ the wreath product of the group of order 2 and the symmetric group $S_n\text{.}$ It has order $2^{n}n!\text{.}$
(c)	The group of type $D_n$ is a subgroup of index 2 in the Coxeter group of type $B_n\text{.}$
(d)	The group of type $I_2\left(m\right)$ is a dihedral group of order $2m\text{.}$

A finite group $W$ generated by reflections in ${ℝ}^n$ is crystallographic if there is a lattice in ${ℝ}^n$ which is stable under the action of $W\text{.}$ The crystallographic finite Coxeter groups are also called Weyl groups. The irreducible Weyl groups are the irreducible finite Coxeter groups of types $$A_{n-1},B_n,D_n,E_6,E_7,E_8,F_4,G_2=I_2\left(6\right)\text{.}$$

References

The most comprehensive reference for finite groups generated by reflections is [Bou1968]. See also the book of Humphreys [Hum1990].

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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