Abstract Root Systems

Root System

A root system is a finite-dimensional real vector space E with an inner product <⋅,⋅>, together with a finite collection R of nonzero vectors in E satisfying the following properties:

1. The vectors in R span E.

2. If α is in R, then so is -α.

3. If α is in R, then the only multiples of α in R are α and -α.

4. If α and β are in R, then so is w_α⋅β, where w_α is the linear transformation of E defined by w_α⋅β = β - 2(<β,α>/<α,α>)α,  β ∈ E. Note: w_α⋅α = -α.

5. For all α and β in R, the quantity 2<β,α>/<α,α> is an integer.

The map w_α is the reflection about the hyperplane perpendicular to α; that is, w_α⋅α = -α and w_α⋅β = β for all β in E that are perpendicular to α. It should be evident that  w_α is an orthogonal transformation of E with determinant -1. 

Since the orthogonal projection of β onto α is given by (<β,α>/<α,α>)α, we note that the quantity 2<β,α>/<α,α> is twice that the coefficient of α in this projection. Thus, the it is equivalent to saying that the projection of β onto α is an integer or half-integer multiple of α. 

Suppose (E,R) and (F,S) are root systems. Consider the vector space E ⊕ F, with the natural inner product determined by the inner products on E and F. Then, R ∪ S is a root system in E ⊕ F, called the direct sum of R and S. 

A root system (E,R) is called reducible if there exists an orthogonal decomposition E = E₁⊕ E₂ with dim E₁ > 0 and dim E₂ > 0 such that every element in R is either in E₁ or in E₂. If no such decomposition exists, (E,R) is called irreducible.

Two root systems (E,R) and (F,S) are said to be equivalent if there exists an invertible linear transformation A : E --> F such that A maps R onto S and such that for all α ∈ R and β ∈ E, we have A(w_α⋅β ) = w_α⋅Aβ. A map A with this property is called an equivalence. Note that the linear map A is not required to preserve inner products, but only to preserve the reflections about the roots. 

Rank

The dimension of E is called the rank of the root system and the elements of R are called roots.

The A₁ rank-one root system R must consist of a pair {α, -α}, where α is a nonzero element of E. 

In rank-two root system, there are four possibilities: A₁ x A₁, A₂, B₂, and G₂. In A₁ x A₁, the lengths of the horizontal roots are unrelated to the lengths of the vertical roots. In A₂, all roots have the same length; angle between successive roots is 60º. In B₂, the length of the longer roots is √2 times the length of the shorter roots;  angle between successive roots is 45º. In G₂, the length of the longer roots is √3 times the length of the shorter roots;  angle between successive roots is 30º. 

Weyl Group

If (E, R) is a root system, then the Weyl group W of R is the subgroup of the orthogonal group of E generated by the reflections w_α, α ∈ R. By assumption, each w_α maps R into itself, indeed onto itself, since each α ∈ R satisfies α = w_α⋅ (w_α⋅α). It follows that every element of W maps R onto itself.

Since the roots span E, a linear transformation of E is determined by its action on R.  This shows that the Weyl group is a finite subgroup of O(E) and may be regarded as a subgroup of the permutation group on R.