# nLab weight (in representation theory)

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

# Contents

## 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

## References

Last revised on March 29, 2014 at 09:02:28. See the history of this page for a list of all contributions to it.