nLab
weight (in representation theory)

Context

Representation theory

Ingredients

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Definitions

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Geometric representation theory

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Theorems

Contents

Definition

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

Definition

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

Definition

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

Remark

In other words, the weight space of a GG-representation for a weight α\alpha is the corresponding eigenspace under the action of TT.

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.