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. 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. Let Φ {displaystyle Phi } be a root system in a Euclidean space V {displaystyle V} . For each root α ∈ Φ {displaystyle alpha in Phi } , let s α {displaystyle s_{alpha }} denote the reflection about the hyperplane perpendicular to α {displaystyle alpha } , which is given explicitly as where ( ⋅ , ⋅ ) {displaystyle (cdot ,cdot )} is the inner product on V {displaystyle V} . The Weyl group W {displaystyle W} of Φ {displaystyle Phi } is the subgroup of the orthogonal group O ( V ) {displaystyle O(V)} generated by all the s α {displaystyle s_{alpha }} 's. By the definition of a root system, each s α {displaystyle s_{alpha }} preserves Φ {displaystyle Phi } , from which it follows that W {displaystyle W} is a finite group. In the case of the A 2 {displaystyle A_{2}} 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, W {displaystyle W} 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, W {displaystyle W} is not the full symmetry group of the root system; a 60-degree rotation preserves Φ {displaystyle Phi } but is not an element of W {displaystyle W} . We may consider also the A n {displaystyle A_{n}} root system. In this case, V {displaystyle V} is the space of all vectors in R n + 1 {displaystyle mathbb {R} ^{n+1}} whose entries sum to zero. The roots consist of the vectors of the form e i − e j , i ≠ j {displaystyle e_{i}-e_{j},,i eq j} , where e i {displaystyle e_{i}} is the i {displaystyle i} th standard basis element for R n + 1 {displaystyle mathbb {R} ^{n+1}} . The reflection associated to such a root is the transformation of V {displaystyle V} obtained by interchanging the i {displaystyle i} th and j {displaystyle j} th entries of each vector. The Weyl group for A n {displaystyle A_{n}} is then the permutation group on n + 1 {displaystyle n+1} elements. If Φ ⊂ V {displaystyle Phi subset V} is a root system, we may consider the hyperplane perpendicular to each root α {displaystyle alpha } . Recall that σ α {displaystyle sigma _{alpha }} denotes the reflection about the hyperplane and that the Weyl group is the group of transformations of V {displaystyle V} generated by all the σ α {displaystyle sigma _{alpha }} '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 v ∈ V {displaystyle vin V} such that ( α , v ) > 0 {displaystyle (alpha ,v)>0} for all α ∈ Δ {displaystyle alpha in Delta } . Since the reflections σ α , α ∈ Φ {displaystyle sigma _{alpha },,alpha in Phi } preserve Φ {displaystyle Phi } , they also preserve the set of hyperplanes perpendicular to the roots. Thus, each Weyl group element permutes the Weyl chambers.