I'm looking for a book or introductory article, that explains branching rules in representation theory of real Lie groups. When a Lie group has a set of irreducible representations, I'd like to know how these representations decompose into irreducible representations of a subgroup. I heard of "Symmetry, representations, and invariants" by Goodman and Wallach and "Representation Theory" by Fulton and Harris, but I couldn't get an account on the special cases I'm interested in, which are $U(1) \to SU(2)$ and $SO(4) \to SO(5)$. I know that $U(1) \cong \operatorname{Spin}(2, \mathbb{R})$ and $SU(2) \cong \operatorname{Spin}(3, \mathbb{R})$, but Goodman/Wallach and Fulton/Harris only seem to treat $\operatorname{Spin}(n, \mathbb{C})$ and $SO(n, \mathbb{C})$.
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4$\begingroup$ $\text{U}(1)$ is the maximal torus of $\text{SU}(2)$, so branching here is completely described by the weight decomposition, which any textbook on the subject surely discusses. $\endgroup$– Qiaochu YuanCommented Nov 14, 2013 at 19:12
3 Answers
Zhelobenko has books on the subject from 1970, 1983, 1994, 2004. I'm pretty sure it's the 1970 that I saw the most concrete branching laws in.
The special cases you're interested in are really easy, by the way, in that the branching laws are "multiplicity-free". For example, the $SU(2)$ irrep of dimension $n+1$ breaks into the $U(1)$ irreps with weights $n,n-2,n-4,\ldots,-n$, each only once.
Though I'm far from being a hands-on expert on the subject, I do know that there are several different approaches to branching rules. As you've seen, Goodman and Wallach have a reasonable (but mainly algebraic) theoretical treatment. There are also expositions geared more explicitly toward compact Lie groups, as in Daniel Bump's Springer GTM 225 Lie Groups (which is apparently about to be published in an expanded second edition). Another alternative is the mathematical physics literature, including older and newer textbooks.
Tables have become rather outmoded now, but a couple of Canadian mathematical physicists long ago published tables of branching rules which might easily answer your specific questions (or not):
Jiri Patera and David Sankoff, Tables of branching rules for representations of simple Lie algebras. Les Presses de l’Universit´e de Montreal, Montreal, Que., 1973. 99 pp.
The underlying mathematics, usually based on Weyl's formulas and Schur-Weyl duality, is well understood by now, but it's not clear to me what user-friendly computer programs are available to avoid excess effort in computing specific branchings.
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$\begingroup$ Sage has a library for this: sagemath.org/doc/thematic_tutorials/lie/… But since I don't understand much of branching rules yet, I couldn't appreciate this much. $\endgroup$ Commented Nov 15, 2013 at 19:06
T. Kobayashi (recently joint with Birgit Speh), has written many papers giving details about branching for classical groups/algebras. Just "google" "T. Kobayashi, branching laws". Much of it is on-line.
