7 - Groups Generated by Pseudoreflections

Summary

Reflections and pseudoreflections

Suppose that V is a finite dimensional vector space over a field K. Recall from Section 2.6 that a pseudoreflection is a linear automorphism of V of finite order whose fixed points have codimension one. If the automorphism is diagonalizable, this is the same as saying that all but one of the eigenvalues are equal to one. A reflection is a diagonalizable pseudoreflection of order two. In this case, the remaining eigenvalue is equal to -1.

If G₁ and G₂ are generated by pseudoreflections on vector spaces V₁ and V₂ respectively, then G₁ × G₂ is generated by pseudoreflections on the space V₁ ⊕ V₂. If G is generated by pseudoreflections on V and cannot be decomposed in this way, then G is said to be an indecomposable pseudoreflection group.

If K = ℚ, the field of rational numbers, then all pseudoreflections are reflections. The product of two distinct reflections is a planar rotation, and since its trace is rational, this product has order two, three, four or six. The finite groups generated by reflections in this case are called the crystallographic reflection groups. The indecomposable ones are in one–one correspondence with the Dynkin diagrams A_n, B_n, D_n, E₆, E₇, E₈, F₄ and G₂ (the reflection groups corresponding to C_n are the same as those for B_n; only the root lengths are different). For a further discussion of the crystallographic reflection groups, see Humphreys [50], Chapter III, and Bourbaki [17].