Coxeter group

A Coxeter groupMathworldPlanetmath G is a finitely generated group, which carries a presentationMathworldPlanetmathPlanetmathPlanetmath of the form


where the integers mij satisfy mii=1 for i=1,,n and mij=mji2 for ij. The exponents form a matrix [mij]1i,jn often called the Coxeter matrix. This is a cousin of the Cartan matrixMathworldPlanetmath and both encode the information of the Dynkin diagramsMathworldPlanetmath.

A Dynkin diagram is the graph with the adjacency matrixMathworldPlanetmath given by [mij-2]1i,jn where [mij]1i,jn is a Coxeter matrix.

A finite Coxeter group is irreduciblePlanetmathPlanetmath if it is not the direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of smaller coxeter groups. These groups are classified and labeled labeled by the Bourbaki types


The classification depends on realizing the groups as reflectionsMathworldPlanetmathPlanetmath of hyperplanesPlanetmathPlanetmath in a finite dimensional real vector space. Then observing a condition on an inner product to be integer valued, it is possible to show these families of symmetryPlanetmathPlanetmath are all that can exist. The Cartan matrix encodes these integer values of the inner product of adjacent reflections while the Coxeter matrix encodes the orders of adjacent products of generatorsPlanetmathPlanetmathPlanetmath.

Remark 1.

The notation An should not be confused with the natation for the alternating groupMathworldPlanetmath on n elements, An. This unfortunate overlap is also a problem with Dn which is not the same as the dihedral groupMathworldPlanetmath on n-vertices, Dn.

Alternative methods to study Coxeter groups is through the use of a length measurement on elements in the group. As every element in g in a Coxeter group is the product of the involutionsPlanetmathPlanetmathPlanetmath w1,,wn, the length is defined as the shortest word in these wis to equal g. We denote this ł(g). Then using careful analysis and the exchange condition it is also possible to specify many of the necessary properties of irreducible Coxeter groups.

Recall that a Weyl groupMathworldPlanetmath W is a group generated by involutions S, that is, generated by elements of order 2. The exchange condition on a W with respect to S states that given a reduced word w=wi1wik in W, wiS, such that for every sS, ł(sw)ł(w) then there exists an j such that


The insistence that wi2=1 shows that Coxeter groups are generated by involutions. This makes every Coxeter group a Weyl group. However, not every Weyl group is a Coxeter group.

The remaining condition to make a Weyl group a Coxeter group is the exchange condition. Thus every finite Weyl group with the exchange condition is a Coxeter group, and visa-versa.

Coxeter groups arrise as the Weyl groups of Lie algebraMathworldPlanetmath, Lie groups, and groups of with a BN-pair. However many other usese exist. It should be noted that the study of Lie theory makes use only of the crystallographic coxeter groups, which are those of type


Thus it omits 𝖧2n, 𝖨3 and 𝖨4

1 Coxeter groups as reflections

Let us see more concretely how a finite Coxeter group can be realized.

Let V be a real Euclidean vector space and 𝒪(V) the group of all orthogonal transformationsMathworldPlanetmath of V.

A reflection of V is a linear transformation S that carries each vector to its mirror image with respect to a fixed hyperplane 𝒫; it is clear geometrically that a reflection is an orthogonal transformation.

A subgroupMathworldPlanetmathPlanetmath 𝒢𝒪(V) will be called effective if V0(𝒢)=0 where V0(𝒢)=T𝒢{xVTx=x}.

A finite Coxeter group can be realized as (i.e. is always isomorphicPlanetmathPlanetmathPlanetmathPlanetmath to) a finite effective subgroup 𝒢 of 𝒪(V) that is generated by a set of reflections, for some Euclidean space V.

2 Classification of irreducible finite Coxeter groups

Type 𝖠n: This group is isomorphic to the symmetric groupMathworldPlanetmathPlanetmath on n elements, Sn. The coxeter matrix is encoded by mi,i+1=3=mi+1,i and all other terms are 2. To observe the isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath let


Then wi2=1, for instance (1,2)2=(), (wiwj)2=1 if |i-j|>1, for example ((1,2)(3,4))2=0 and (wiwi+1)3=1 as we see with (1,2)(2,3)=(1,2,3) which has order 3.

The Dynkin diagram is:


Type 𝖡n, 𝖢n: This group is isomorphic to the wreath product 2Sn, that is, the semi-direct product of 2nSn where Sn permutes the entries of the vectors in 2n.

The designation of type 𝖡n and 𝖢n relate to the fact that two different methods can be given to construct the same group (as the Weyl group of O(2n+1,k) or as the Weyl group of Sp(2n,k)). It is also common to see 𝖢n used as the sole label.

Type 𝖧2n: These groups are the dihedral group D2n for n5 and n6.

Type 𝖦2: This group is isomorphic to S3.


L. C. Grove, C. T. Benson, Finite Reflection Groups. Second Edition., Springer-Verlag, 1985.

Title Coxeter group
Canonical name CoxeterGroup
Date of creation 2013-03-22 15:38:20
Last modified on 2013-03-22 15:38:20
Owner Simone (5904)
Last modified by Simone (5904)
Numerical id 12
Author Simone (5904)
Entry type Definition
Classification msc 20F55
Defines Coxeter group
Defines Coxeter matrix
Defines Cartan matrix
Defines Weyl group
Defines exchange condition
Defines length of Weyl group