A Dynkin diagram is the graph with the adjacency matrix given by where is a Coxeter matrix.
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.
Alternative methods to study Coxeter groups is through the use of a length measurement on elements in the group. As every element in in a Coxeter group is the product of the involutions , the length is defined as the shortest word in these to equal . We denote this . 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 is a group generated by involutions , that is, generated by elements of order 2. The exchange condition on a with respect to states that given a reduced word in , , such that for every , then there exists an such that
The insistence that 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
Thus it omits , and
1 Coxeter groups as reflections
Let us see more concretely how a finite Coxeter group can be realized.
A reflection of is a linear transformation that carries each vector to its mirror image with respect to a fixed hyperplane ; it is clear geometrically that a reflection is an orthogonal transformation.
2 Classification of irreducible finite Coxeter groups
Then , for instance , if , for example and as we see with which has order 3.
The Dynkin diagram is:
The designation of type and relate to the fact that two different methods can be given to construct the same group (as the Weyl group of or as the Weyl group of ). It is also common to see used as the sole label.
Type : These groups are the dihedral group for and .
Type : This group is isomorphic to .
L. C. Grove, C. T. Benson, Finite Reflection Groups. Second Edition., Springer-Verlag, 1985.
|Date of creation||2013-03-22 15:38:20|
|Last modified on||2013-03-22 15:38:20|
|Last modified by||Simone (5904)|
|Defines||length of Weyl group|