Last update: 17 September 2013
A Coxeter group is a group presented by generators and relations where each is either or a positive integer greater than 1.
A reflection is a linear transformation of which is a reflection in some hyperplane.
A finite group generated by reflections is a finite subgroup of 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”|
The irreducible finite Coxeter groups of classical type are the ones of types and and the others are the irreducible finite Coxeter groups of exceptional type.
|(a)||The group of type is the symmetric group|
|(b)||The group of type is the hyperoctahedral group the wreath product of the group of order 2 and the symmetric group It has order|
|(c)||The group of type is a subgroup of index 2 in the Coxeter group of type|
|(d)||The group of type is a dihedral group of order|
The most comprehensive reference for finite groups generated by reflections is [Bou1968]. See also the book of Humphreys [Hum1990].
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..