Weyl group

In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system.

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.

is isomorphic to the permutation group on three elements, which we may think of as the vertices of the triangle.

is not the full symmetry group of the root system; a 60-degree rotation preserves

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

, they also preserve the set of hyperplanes perpendicular to the roots.

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.

A basic general theorem about Weyl chambers is this:[2] A related result is this one:[3] A key result about the Weyl group is this:[4] That is to say, the group generated by the reflections

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.

are orthogonal, from which it follows easily that the corresponding reflections commute.

More generally, the number of bonds determines the angle

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 (xixj)mij=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.

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.

There are also various definitions of Weyl groups specific to various group-theoretic and geometric contexts (Lie algebra, Lie group, symmetric space, etc.).

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.

,[6] at which point one has an alternative description of the Weyl group as Now, one can define a root system

; the roots are the nonzero weights of the adjoint action of

[7] With a bit more effort, one can show that these reflections generate all of

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.

) 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.

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.[5] If B is a Borel subgroup of G, i.e., a maximal connected solvable subgroup and a maximal torus T = T0 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.

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.

as:[8] The outer automorphisms of the group Out(G) are essentially the diagram automorphisms of the Dynkin diagram, while the group cohomology is computed in Hämmerli, Matthey & Suter 2004 and is a finite elementary abelian 2-group (