Branching rule for Lie representations

Definition

Let $i:H\to G$ be a homomorphism of Lie groups and let $(\pi,V)$ be a finite-dimensional representation of $G$ . The branching rule for $V$ along $i$ is the decomposition of the restricted representation

\operatorname{Res}^{G}_{H}V, \qquad h\longmapsto \pi(i(h)),

into irreducible $H$ -representations. When restriction is completely reducible, it has the form

\operatorname{Res}^{G}_{H}V\cong \bigoplus_{\mu}m_{\mu}W_{\mu},

and the branching rule records the occurring irreducibles $W_\mu$ and their multiplicities $m_\mu$ .

The same terminology applies to a homomorphism of Lie algebras $\mathfrak h\to\mathfrak g$ : one restricts a $\mathfrak g$ -representation to $\mathfrak h$ and decomposes it there.

Typical settings

Complete reducibility holds, for example, for finite-dimensional representations of compact groups and for finite-dimensional representations of complex semisimple Lie algebras. For a complex reductive Lie algebra, one must additionally require the center to act semisimply. In a highest-weight setting a branching rule is often presented as a formula

V_\lambda\big|_H\cong\bigoplus_\mu m_{\lambda\mu}W_\mu

relating the highest-weight labels for $G$ and $H$ .

Restriction may be computed by comparing characters, by decomposing weight spaces, or by using invariant tensors. A multiplicity-free branching rule is one for which every coefficient $m_{\lambda\mu}$ is at most one.

Example

For the standard inclusion $SO(n-1)\subset SO(n)$ , the defining $n$ -dimensional representation restricts as

\mathbb R^n\big|_{SO(n-1)}\cong \mathbb R^{n-1}\oplus\mathbb R,

where the last summand is the trivial representation carried by the fixed coordinate axis.

Global and infinitesimal cautions

A Lie-group branching rule differentiates to the corresponding Lie-algebra restriction, but the converse need not determine the group-level rule. A Lie algebra cannot see disconnected components, and a representation of a Lie algebra may integrate only to a covering group rather than to the chosen global form of $G$ . Central quotients can therefore remove representations that are allowed infinitesimally.

If the restricted representation is not semisimple, a direct-sum branching formula may not exist. One must then specify whether the desired data are composition-factor multiplicities, a filtration, or a decomposition into indecomposable modules.

References

Roe Goodman and Nolan R. Wallach, Symmetry, Representations, and Invariants, Springer, 2009, Chapters 5 and 8. Publisher record.
William Fulton and Joe Harris, Representation Theory: A First Course, Springer, 1991, §§8 and 25. Publisher record.