Orthogonal reflections

A [Euclidean space] is a finite-dimensional real vector space V equipped with an [inner product] (-, -) \colon V \times V \to \bbR (a symmetric positive-definite bilinear form). Given any hyperplane H \subseteq V and vector v, the inner product defines an [orthogonal reflection] of v over H, by travelling from v to H along the shortest path, and then travelling that distance again. The inner product can also define the hyperplane H as the kernel of the map v \mapsto (\alpha, v) for some fixed vector \alpha \in V, in which case we use the notation H_\alpha for the hyperplane.

The following diagram shows the situation in two dimensions, with the vector \alpha defining the hyperplane H_\alpha, and s_\alpha \colon V \to V the reflection-over-H_\alpha operator. [Try moving \alpha and v and see how the diagram changes.]

As illustrated in the diagram, the formula for the reflection s_\alpha is s_\alpha(v) = v - 2 \frac{(v, \alpha)}{(\alpha, \alpha)}.

The rank 1 and 2 root systems

A Euclidean root system is a finite set \Phi of nonzero vectors inside a Euclidean space (V, (-, -)), which satisfy the two additional properties:

1. ([Reflection invariance]) We have s_\alpha(\Phi) = \Phi for all \alpha \in \Phi, and
2. ([Crystallographic]) For every pair \alpha, \beta \in \Phi, the number 2 \frac{(\alpha, \beta)}{\alpha, \alpha} is an integer.

The vectors in \Phi are called [roots], while \dim (\span_\bbR \Phi) is called the [rank] of the root system.

There are exactly two rank-1 root systems. The A_1 root system consists of \{\pm \alpha\} for any nonzero vector \alpha \in V, and the BC_1 root system consists of \{\pm \alpha, \pm 2 \alpha\}.

A simple application of the crystallographic property shows that if \alpha is a root, then the only multiples of \alpha which could also be roots are \pm \frac{1}{2} \alpha, \pm \alpha, or \pm 2 \alpha, and so indeed BC_1 is as complicated as a rank 1 system could get. We call a root \alpha \in \Phi [divisible] if \frac{1}{2} \alpha is also a root, and [indivisible] otherwise. The root system A_1 is called [reduced], meaning that every root is indivisible, while the root system BC_1 is [non-reduced].

Rank 2 root systems are where the geometry starts getting interesting. There are seven kinds of rank 2 root systems, four of which are reduced. All of these except for A_2 and G_2 are constructed in the Euclidean space \bbR^2, while A_2 and G_2 are constructed in the plane Z \subseteq \bbR^3 defined by x_1 + x_2 + x_3 = 0. [Move your mouse over the roots to see their coordinates.]

The root systems A_1 \times A_1, A_1 \times BC_1, and BC_1 \times BC_1 are called [reducible] because we can decompose the ambient space into two orthogonal subspaces, each containing a part of the root system. The other root systems A_2, B_2, G_2 and BC_2 are called [irreducible] since they cannot be decomposed in this way. (Be careful not to confuse “reduced” with “irreducible”). The rank 2 root systems which are both reduced and irreducible are A_2, B_2, and G_2.

Building a root system

It is not difficult to prove that the only rank 2 root systems are those shown above. In order to give some intuition for this problem, here is a game where the goal is to build a rank 2 root system.

Positive and simple systems

[Planned for this section:]

1. Show and explain gridlines, root lattice, weight lattice.
2. Positive and simple systems
3. Coxeter and Cartan matrices