Geometry MOC

Root system

A root system Φ is a finite set of vectors, called roots, in the quadratic space 𝔼 =ℚ𝑛,0 such that geo

1. Φ spans 𝔼;
2. for every root 𝛼 ∈Φ, the reflection 𝜎𝛼 through the hyperplane¹ perpendicular to 𝛼 leaves Φ invariant.

Often one also requires

3. (reduced root system) if 𝛼,𝛽 ∈Φ and 𝛽 ∝𝛼, then 𝛽 = ±𝛼
4. (crystallographic root system) if 𝛼,𝛽 ∈Φ, the projection of 𝛼 onto 𝛽 is an integer or half-integer multiple of 𝛽

We will call a reduced crystallographic root system an RC root system. A root system which is not necessarily RC will sometimes be called a general root system for emphasis. Denoting the bilinear product on 𝔼 as ( ⋅, ⋅) we define

⟨𝛽,𝛼⟩=2(𝛽,𝛼)(𝛼,𝛼)

which is linear in 𝛽 only, whence

𝜎𝛼(𝛽)=𝛽−⟨𝛽,𝛼⟩𝛼

and we may reëxpress ^R4 as²

4. (crystallographic root system) if 𝛼,𝛽 ∈Φ, then ⟨𝛽,𝛼⟩ ∈ℤ.

Further notions

• An isomorphism 𝜑 :Φ →Φ′ of root systems is an isometry 𝜑 ∈O⁡(𝔼) of 𝔼 such that 𝜑(Φ) =Φ′.
• The subgroup of automorphisms generated by reflections 𝜎𝛼 is called its Weyl group.
• Dual root system

Properties

1. Reflections of a general root system
2. Conjugation of a Weyl element

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develop | en | SemBr

Footnotes

1. i.e. subspace of codimension 1 ↩

2. 1972. Introduction to Lie Algebras and Representation Theory, §9.1–9.2, pp. 42–43 ↩