Coxeter group

A Coxeter group $G$ is a finitely generated group, which carries a presentation of the form

W=\\langle w_{1},\\dots,w_{n}\\mid(w_{i}w_{j})^{m_{ij}}=1\\rangle

where the integers $m_{ij}$ satisfy $m_{ii}=1$ for $i=1,\\dots,n$ and $m_{ij}=m_{ji}\\geq 2$ for $i\eq j$ . The exponents form a matrix $[m_{ij}]_{1\\leq i,j\\leq n}$ often called the Coxeter matrix.This is a cousin of the Cartan matrix and both encode the informationof the Dynkin diagrams.

A Dynkin diagram is the graph with the adjacency matrix given by $[m_{ij}-2]_{1\\leq i,j\\leq n}$ where $[m_{ij}]_{1\\leq i,j\\leq n}$ is aCoxeter matrix.

A finite Coxeter group is irreducible if it is not the direct productof smaller coxeter groups. These groups are classified and labeledlabeled by the Bourbaki types

\\mathsf{A}_{n},\\mathsf{B}_{n},\\mathsf{C}_{n},\\mathsf{D}_{n},\\mathsf{E}_{6},%\\mathsf{E}_{7},\\mathsf{E}_{8},\\mathsf{F}_{4},\\mathsf{G}_{2},\\mathsf{H}_{2}^{n}%,\\mathsf{I}_{3},\\mathsf{I}_{4}.

The classification depends on realizing the groups asreflections of hyperplanes in a finite dimensional real vector space.Then observing a condition on an inner product to be integer valued, itis possible to show these families of symmetry are all that can exist.The Cartan matrix encodes these integer values of the inner productof adjacent reflections while the Coxeter matrix encodes the orders ofadjacent products of generators.

Remark 1.

The notation $\\mathsf{A}_{n}$ should not be confused with the natation forthe alternating group on $n$ elements, $A_{n}$ . This unfortunate overlap isalso a problem with $\\mathsf{D}_{n}$ which is not the same as the dihedralgroup on $n$ -vertices, $D_{n}$ .

Alternative methods to study Coxeter groups is through the use of a lengthmeasurement on elements in the group. As every element in $g$ in a Coxetergroup is the product of the involutions $w_{1},\\dots,w_{n}$ , the length is defined as the shortest word in these $w_{i}^{\\prime}s$ to equal $g$ . We denote this $\\l(g)$ . Then using carefulanalysis and the exchange condition it is also possible to specify many of the necessary properties of irreducible Coxeter groups.

Recall that a Weyl group $W$ 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 $w=w_{i_{1}}\\cdots w_{i_{k}}$ in $W$ , $w_{i}\\in S$ , such that for every $s\\in S$ , $\\l(sw)\\leq\\l(w)$ then there exists an $j$ such that

sw=w_{i_{1}}\\cdots w_{i_{j-1}}w_{i_{j+1}}\\cdots w_{i_{k}}.

The insistence that $w_{i}^{2}=1$ shows that Coxeter groups are generated byinvolutions. 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 aCoxeter group, and visa-versa.

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

\\mathsf{A}_{n},\\mathsf{B}_{n},\\mathsf{C}_{n},\\mathsf{D}_{n},\\mathsf{E}_{6},%\\mathsf{E}_{7},\\mathsf{E}_{8},\\mathsf{F}_{4},\\mathsf{G}_{2}.

Thus it omits $\\mathsf{H}_{2}^{n}$ , $\\mathsf{I}_{3}$ and $\\mathsf{I}_{4}$

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 $\\mathcal{O}(V)$ the group of all orthogonal transformations 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 $\\mathcal{P}$ ; it is clear geometrically that a reflection is an orthogonal transformation.

A subgroup $\\mathcal{G}\\leq\\mathcal{O}(V)$ will be called effective if $V_{0}(\\mathcal{G})=0$ where $V_{0}(\\mathcal{G})=\\bigcap_{T\\in\\mathcal{G}}\\{x\\in V\\mid Tx=x\\}$ .

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

2 Classification of irreducible finite Coxeter groups

Type $\\mathsf{A}_{n}$ : This group is isomorphic to the symmetric group on $n$ elements, $S_{n}$ . The coxeter matrix is encoded by $m_{i,i+1}=3=m_{i+1,i}$ and all other terms are 2. To observe the isomorphism let

w_{1}=(1,2),(2,3),\\dots,w_{n}=(n-1,n).

Then $w_{i}^{2}=1$ , for instance $(1,2)^{2}=()$ , $(w_{i}w_{j})^{2}=1$ if $|i-j|>1$ , for example $((1,2)(3,4))^{2}=0$ and $(w_{i}w_{i+1})^{3}=1$ as we see with $(1,2)(2,3)=(1,2,3)$ which has order 3.

The Dynkin diagram is:

\\xymatrix{\\circ\\ar@{-}[r]&\\circ\\ar@{-}[r]&\\circ\\ar@{--}[r]&\\circ.}

Type $\\mathsf{B}_{n}$ , $\\mathsf{C}_{n}$ : This group is isomorphic to the wreath product $\\mathbb{Z}_{2}\\wr S_{n}$ , that is, the semi-direct product of $\\mathbb{Z}_{2}^{n}\\rtimes S_{n}$ where $S_{n}$ permutes the entries of the vectors in $\\mathbb{Z}_{2}^{n}$ .

The designation of type $\\mathsf{B}_{n}$ and $\\mathsf{C}_{n}$ 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 $\\mathsf{C}_{n}$ used as the sole label.

Type $\\mathsf{H}_{2}^{n}$ : These groups are the dihedral group $D_{2n}$ for $n\\geq 5$ and $n\eq 6$ .

Type $\\mathsf{G}_{2}$ : This group is isomorphic to $S_{3}$ .

References

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