root (in representation theory)




Let EE be a finite-dimensional Euclidean vector space, i.e. a real vector space equipped with a positive-definite inner product |\langle | \rangle. For convenience, given any vectors v,wEv, w \in E with w0w \neq 0, we define the notation vw2v|ww|wv \dashv w \coloneqq 2 \frac{\langle v | w \rangle}{\langle w | w \rangle}. Notice that this is precisely twice the coefficient of ww when we take the orthogonal projection of vv onto the line generated by ww. We denote by σ w\sigma_w the reflection in the hyperplane P w{w} P_w \coloneqq \{ w \}^{\perp}, which is given by the formula

σ w(v)=v(vw)w\sigma_w(v) = v - (v\dashv w)w

A root system in EE is a finite set of nonzero vectors ΔE{0}\Delta \subset E \setminus \{0 \} satisfying the following axioms:

  1. Δ\Delta spans EE.

  2. If αΔ\alpha \in \Delta, then the reflection σ α\sigma_\alpha permutes Δ\Delta.

  3. (Integrality) If α,βΔ\alpha, \beta \in \Delta then αβ\alpha \dashv \beta \in \mathbb{Z}.

If in addition for every αΔ\alpha \in \Delta we have span(α)Δ={α,α},\operatorname{span}(\alpha) \cap \Delta = \{- \alpha, \alpha \}, then we say that the root system is reduced. Equivalently, a root system is reduced if each αΔ\alpha \in \Delta determines a unique reflection. Many authors require that root systems are reduced, but we will specify here when root systems are reduced or not.

Weyl group

The Weyl group of a root system Δ\Delta in EE is the finite subgroup W(Δ)W(\Delta) of the orthogonal group O(E)\operatorname{O}(E) generate by the reflections σ α\sigma_\alpha, αΔ\alpha \in \Delta. Thus, the Weyl group can also be seen as a subgroup of the permutation group of Δ\Delta.

Category of root systems

Given root systems Δ\Delta in EE and Δ\Delta' in EE', we define a morphism of root systems from Δ\Delta to Δ\Delta' to be a linear map f:EEf: E \to E' such that f(Δ)Δf(\Delta) \subseteq \Delta' and f(α)f(β)=αβf(\alpha) \dashv f(\beta) = \alpha \dashv \beta for all α,βΔ\alpha, \beta \in \Delta.


  • Peter Woit, Topics in representation theory: Roots and weights (pdf)

  • The Unapologetic Mathematician, The category of root systems (blog)

Last revised on May 8, 2021 at 14:53:15. See the history of this page for a list of all contributions to it.