nLab weight (in representation theory)

Definition

Let $G$ be a Lie group which is compact and connected. Write $T \hookrightarrow G$ for the maximal torus subgroup.

Definition

A weight on $G$ is an irreducible representation of the maximal torus subgroup $T \hookrightarrow G$.

Definition

For $\rho : G \to Aut(V)$ a representation of $G$, and for $\alpha : T \to Aut(\mathbb{C})$ a weight, the weight space of $\rho$ with respect to $\alpha$ is the subspace of $V$ which as a representation of $T$ is a direct sum of $\alpha$-s.

Remark

In other words, the weight space of a $G$-representation for a weight $\alpha$ is the corresponding eigenspace under the action of $T$.

Properties

For connected compact Lie groups

For connected compact Lie groups the

Related concepts

• group character

• root (in representation theory)

• maximal torus, Cartan subalgebra

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)

• Wikipedia, Weight (representation theory)