Weyl group

In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection group. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these.
The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linear algebraic group, etc. is the Weyl group of the root system of that group or algebra.
It is named after Hermann Weyl.
Definition and examples.
Let formula_1 be a root system in a Euclidean space formula_2. For each root formula_3, let formula_4 denote the reflection about the hyperplane perpendicular to formula_5, which is given explicitly as
where formula_7 is the inner product on formula_2. The Weyl group formula_9 of formula_1 is the subgroup of the orthogonal group formula_11 generated by all the formula_4's. By the definition of a root system, each formula_4 preserves formula_1, from which it follows that formula_9 is a finite group.
In the case of the formula_16 root system, for example, the hyperplanes perpendicular to the roots are just lines, and the Weyl group is the symmetry group of an equilateral triangle, as indicated in the figure. As a group, formula_9 is isomorphic to the permutation group on three elements, which we may think of as the vertices of the triangle. Note that in this case, formula_9 is not the full symmetry group of the root system; a 60-degree rotation preserves formula_1 but is not an element of formula_9.
We may consider also the formula_21 root system. In this case, formula_2 is the space of all vectors in formula_23 whose entries sum to zero. The roots consist of the vectors of the form formula_24, where formula_25 is the formula_26th standard basis element for formula_23. The reflection associated to such a root is the transformation of formula_2 obtained by interchanging the formula_26th and formula_30th entries of each vector. The Weyl group for formula_21 is then the permutation group on formula_32 elements.
Weyl chambers.
If formula_33 is a root system, we may consider the hyperplane perpendicular to each root formula_5. Recall that formula_4 denotes the reflection about the hyperplane and that the Weyl group is the group of transformations of formula_2 generated by all the formula_4's. The complement of the set of hyperplanes is disconnected, and each connected component is called a Weyl chamber. If we have fixed a particular set Δ of simple roots, we may define the fundamental Weyl chamber associated to Δ as the set of points formula_38 such that formula_39 for all formula_40.
Since the reflections formula_41 preserve formula_1, they also preserve the set of hyperplanes perpendicular to the roots. Thus, each Weyl group element permutes the Weyl chambers.
The figure illustrates the case of the A2 root system. The "hyperplanes" (in this case, one dimensional) orthogonal to the roots are indicated by dashed lines. The six 60-degree sectors are the Weyl chambers and the shaded region is the fundamental Weyl chamber associated to the indicated base.
A basic general theorem about Weyl chambers is this:
A related result is this one:
Coxeter group structure.
Generating set.
A key result about the Weyl group is this:
That is to say, the group generated by the reflections formula_53 is the same as the group generated by the reflections formula_41.
Relations.
Meanwhile, if formula_5 and formula_56 are in formula_48, then the Dynkin diagram for formula_1 relative to the base formula_48 tells us something about how the pair formula_60 behaves. Specifically, suppose formula_45 and formula_62 are the corresponding vertices in the Dynkin diagram. Then we have the following results:
The preceding claim is not hard to verify, if we simply remember what the Dynkin diagram tells us about the angle between each pair of roots. If, for example, there is no bond between the two vertices, then formula_5 and formula_56 are orthogonal, from which it follows easily that the corresponding reflections commute. More generally, the number of bonds determines the angle formula_81 between the roots. The product of the two reflections is then a rotation by angle formula_82 in the plane spanned by formula_5 and formula_56, as the reader may verify, from which the above claim follows easily.
As a Coxeter group.
Weyl groups are examples of finite reflection groups, as they are generated by reflections; the abstract groups (not considered as subgroups of a linear group) are accordingly finite Coxeter groups, which allows them to be classified by their Coxeter–Dynkin diagram. Being a Coxeter group means that a Weyl group has a special kind of presentation in which each generator "xi" is of order two, and the relations other than "xi2=1" are of the form ("x""i""x""j")"m""ij"=1. The generators are the reflections given by simple roots, and "mij" is 2, 3, 4, or 6 depending on whether roots "i" and "j" make an angle of 90, 120, 135, or 150 degrees, i.e., whether in the Dynkin diagram they are unconnected, connected by a simple edge, connected by a double edge, or connected by a triple edge. We have already noted these relations in the bullet points above, but to say that formula_9 is a Coxeter group, we are saying that those are the "only" relations in formula_9.
Weyl groups have a Bruhat order and length function in terms of this presentation: the "length" of a Weyl group element is the length of the shortest word representing that element in terms of these standard generators. There is a unique longest element of a Coxeter group, which is opposite to the identity in the Bruhat order.
Weyl groups in algebraic, group-theoretic, and geometric settings.
Above, the Weyl group was defined as a subgroup of the isometry group of a root system. There are also various definitions of Weyl groups specific to various group-theoretic and geometric contexts (Lie algebra, Lie group, symmetric space, etc.). For each of these ways of defining Weyl groups, it is a (usually nontrivial) theorem that it is a Weyl group in the sense of the definition at the top of this article, namely the Weyl group of some root system associated with the object. A concrete realization of such a Weyl group usually depends on a choice – e.g. of Cartan subalgebra for a Lie algebra, of maximal torus for a Lie group.
The Weyl group of a connected compact Lie group.
Let formula_87 be a connected compact Lie group and let formula_88 be a maximal torus in formula_87. We then introduce the normalizer of formula_88 in formula_87, denoted formula_92 and defined as
We also define the centralizer of formula_88 in formula_87, denoted formula_96 and defined as
The Weyl group formula_9 of formula_87 (relative to the given maximal torus formula_88) is then defined initially as
Eventually, one proves that formula_102, at which point one has an alternative description of the Weyl group as
Now, one can define a root system formula_1 associated to the pair formula_105; the roots are the nonzero weights of the adjoint action of formula_88 on the Lie algebra of formula_87. For each formula_3, one can construct an element formula_109 of formula_92 whose action on formula_88 has the form of reflection. With a bit more effort, one can show that these reflections generate all of formula_112. Thus, in the end, the Weyl group as defined as formula_113 or formula_112 is isomorphic to the Weyl group of the root system formula_1.
In other settings.
For a complex semisimple Lie algebra, the Weyl group is simply "defined" as the reflection group generated by reflections in the roots – the specific realization of the root system depending on a choice of Cartan subalgebra.
For a Lie group "G" satisfying certain conditions, given a torus "T" < "G" (which need not be maximal), the Weyl group "with respect to" that torus is defined as the quotient of the normalizer of the torus "N" = "N"("T") = "NG"("T") by the centralizer of the torus "Z" = "Z"("T") = "ZG"("T"),
The group "W" is finite – "Z" is of finite index in "N". If "T" = "T"0 is a maximal torus (so it equals its own centralizer: formula_117) then the resulting quotient "N"/"Z" = "N"/"T" is called "the" Weyl group of "G", and denoted "W"("G"). Note that the specific quotient set depends on a choice of maximal torus, but the resulting groups are all isomorphic (by an inner automorphism of "G"), since maximal tori are conjugate.
If "G" is compact and connected, and "T" is a "maximal" torus, then the Weyl group of "G" is isomorphic to the Weyl group of its Lie algebra, as discussed above.
For example, for the general linear group "GL," a maximal torus is the subgroup "D" of invertible diagonal matrices, whose normalizer is the generalized permutation matrices (matrices in the form of permutation matrices, but with any non-zero numbers in place of the '1's), and whose Weyl group is the symmetric group. In this case the quotient map "N" → "N"/"T" splits (via the permutation matrices), so the normalizer "N" is a semidirect product of the torus and the Weyl group, and the Weyl group can be expressed as a subgroup of "G". In general this is not always the case – the quotient does not always split, the normalizer "N" is not always the semidirect product of "W" and "Z," and the Weyl group cannot always be realized as a subgroup of "G."
Bruhat decomposition.
If "B" is a Borel subgroup of "G", i.e., a maximal connected solvable subgroup and a maximal torus "T" = "T"0 is chosen to lie in "B", then we obtain the Bruhat decomposition
which gives rise to the decomposition of the flag variety "G"/"B" into Schubert cells (see Grassmannian).
The structure of the Hasse diagram of the group is related geometrically to the cohomology of the manifold (rather, of the real and complex forms of the group), which is constrained by Poincaré duality. Thus algebraic properties of the Weyl group correspond to general topological properties of manifolds. For instance, Poincaré duality gives a pairing between cells in dimension "k" and in dimension "n" - "k" (where "n" is the dimension of a manifold): the bottom (0) dimensional cell corresponds to the identity element of the Weyl group, and the dual top-dimensional cell corresponds to the longest element of a Coxeter group.
Analogy with algebraic groups.
There are a number of analogies between algebraic groups and Weyl groups – for instance, the number of elements of the symmetric group is "n"!, and the number of elements of the general linear group over a finite field is related to the "q"-factorial formula_119; thus the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by the field with one element, which considers Weyl groups to be simple algebraic groups over the field with one element.
Cohomology.
For a non-abelian connected compact Lie group "G," the first group cohomology of the Weyl group "W" with coefficients in the maximal torus "T" used to define it, is related to the outer automorphism group of the normalizer formula_120 as:
The outer automorphisms of the group Out("G") are essentially the diagram automorphisms of the Dynkin diagram, while the group cohomology is computed in and is a finite elementary abelian 2-group (formula_122); for simple Lie groups it has order 1, 2, or 4. The 0th and 2nd group cohomology are also closely related to the normalizer.