nLab root (in representation theory)

Definition

Let $E$ 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 $v, w \in E$ with $w \neq 0$, we define the notation $v \dashv w \coloneqq 2 \frac{\langle v | w \rangle}{\langle w | w \rangle}$. Notice that this is precisely twice the coefficient of $w$ when we take the orthogonal projection of $v$ onto the line generated by $w$. We denote by $\sigma_w$ the reflection in the hyperplane $P_w \coloneqq \{ w \}^{\perp}$, which is given by the formula

$$\sigma_w(v) = v - (v\dashv w)w$$

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

1. $\Delta$ spans $E$.

2. If $\alpha \in \Delta$, then the reflection $\sigma_\alpha$ permutes $\Delta$.

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

If in addition for every $\alpha \in \Delta$ we have $\operatorname{span}(\alpha) \cap \Delta = \{- \alpha, \alpha \},$ then we say that the root system is reduced. Equivalently, a root system is reduced if each $\alpha \in \Delta$ 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 $\Delta$ in $E$ is the finite subgroup $W(\Delta)$ of the orthogonal group $\operatorname{O}(E)$ generate by the reflections $\sigma_\alpha$, $\alpha \in \Delta$. Thus, the Weyl group can also be seen as a subgroup of the permutation group of $\Delta$.

Category of root systems

Given root systems $\Delta$ in $E$ and $\Delta'$ in $E'$, we define a morphism of root systems from $\Delta$ to $\Delta'$ to be a linear map $f: E \to E'$ such that $f(\Delta) \subseteq \Delta'$ and $f(\alpha) \dashv f(\beta) = \alpha \dashv \beta$ for all $\alpha, \beta \in \Delta$.

Related concepts

• weight (in representation theory)

• maximal torus, Cartan subalgebra

• Dynkin diagram

• ADE classification

• Coxeter number

References

• Howard Georgi, §6 in: Lie Algebras In Particle Physics, Westview Press (1999), CRC Press (2019) [doi:10.1201/9780429499210]

  with an eye towards application to (the standard model of) particle physics

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

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