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Coxeter group (Definition)

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

$\displaystyle W=\langle w_1,\dots,w_n\mid (w_iw_j)^{m_{ij}}=1\rangle $
where the integers $ m_{ij}$ satisfy $ m_{ii}=1$ for $ i=1,\dots,n$ and $ m_{ij}=m_{ji}\ge 2$ for $ i\ne 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 information of 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 a Coxeter matrix.

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

$\displaystyle \mathsf{A}_n, \mathsf{B}_n, \mathsf{C}_n, \mathsf{D}_n, \mathsf{E... ...sf{E}_8, \mathsf{F}_4, \mathsf{G}_2, \mathsf{H}_2^n, \mathsf{I}_3,\mathsf{I}_4.$
The classification depends on realizing the groups as reflections of hyperplanes 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 symmetry 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 generators.
Remark 1   The notation $ \mathsf{A}_n$ should not be confused with the natation for the alternating group on $ n$ elements, $ A_n$. This unfortunate overlap is also a problem with $ \mathsf{D}_n$ which is not the same as the dihedral group on $ n$-vertices, $ D_n$.

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 involutions $ w_1,\dots,w_n$, the length is defined as the shortest word in these $ w_i's$ to equal $ g$. We denote this $ \l(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 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

$\displaystyle sw=w_{i_1}\cdots w_{i_{j-1}}w_{i_{j+1}}\cdots w_{i_k}.$

The insistence that $ w_i^2=1$ shows that Coxeter groups are generated 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 algebra, 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

$\displaystyle \mathsf{A}_n, \mathsf{B}_n, \mathsf{C}_n, \mathsf{D}_n, \mathsf{E}_6, \mathsf{E}_7, \mathsf{E}_8, \mathsf{F}_4, \mathsf{G}_2.$
Thus it omits $ \mathsf{H}_2^n$, $ \mathsf{I}_3$ and $ \mathsf{I}_4$

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\le\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$.

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

$\displaystyle w_1=(1,2),(2,3),\dots,w_n=(n-1,n).$
Then $ w_i^2=1$, for instance $ (1,2)^2=()$, $ (w_i w_j)^2=1$ if $ \vert i-j\vert>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:

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ \circ\ar@{-}[r] & \circ\ar@{-}[r] & \circ\ar@{--}[r] & \circ . } } \end{xy}$

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\neq 6$.

Type $ \mathsf{G}_2$: This group is isomorphic to $ S_3$.

Bibliography

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



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Also defines:  Coxeter group, Coxeter matrix, Cartan matrix, Weyl group, exchange condition, length of Weyl group
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Cross-references: label, semi-direct product, wreath product, isomorphism, terms, symmetric group, Euclidean space, isomorphic, effective, subgroup, clear, fixed, mirror image, vector, linear transformation, orthogonal transformations, Euclidean vector space, crystallographic, theory, BN-pair, Lie groups, Lie algebra, reduced word, generated by, group generated by, properties, necessary, word, involutions, length, dihedral group, alternating group, generators, products, orders, adjacent, symmetry, inner product, vector space, real, finite dimensional, hyperplanes, reflections, types, Bourbaki, groups, direct product, irreducible, finite, adjacency matrix, graph, Dynkin diagrams, information, matrix, exponents, integers, presentation, finitely generated group
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This is version 9 of Coxeter group, born on 2006-01-21, modified 2007-04-12.
Object id is 7570, canonical name is CoxeterGroup.
Accessed 3664 times total.

Classification:
AMS MSC: 20F55 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Reflection and Coxeter groups)
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