Geometry MOC

Root system

A root system is a finite set of vectors, called roots, in the quadratic space such that geo

1. spans ;
2. for every root , the reflection through the hyperplane¹ perpendicular to leaves invariant.

Often one also requires

3. (reduced root system) if and , then
4. (crystallographic root system) if , the projection of onto is an integer or half-integer multiple of

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 as we define

which is linear in only, whence

and we may reëxpress ^R4 as²

4. (crystallographic root system) if , then .

Further notions

• An isomorphism of root systems is an isometry of such that .
• The subgroup of automorphisms generated by reflections 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 ↩