nLab restricted representation

Idea

Given a group $G$ and a representation $V$ of $G$, then any subgroup inclusion $H\hookrightarrow G$ makes also $H$ act on $V$, this is the restricted representation.

Given an irreducible representation of $G$, then its decomposition as a direct sum of irreducible representations of $H$, after restricting the respesentation to $H$, is often called the “branching rule” of the restriction.

More generally, for any group homomorphism $H \to G$ representations of $G$ “pull back” to representations of $H$.

The left adjoint to this construction is the operation of forming induced representations, the right adjoint that of coinduced representations.

Examples

• supermultiplet

References

• Wikipedia, Restricted representation