# Where can I learn the rules of representation decompositions? [duplicate]

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I want to understand the rules of how to decompose representations of say $SU(N)$ to simpler representations. For example, in $SU(2)$ the representation $\mathbf{2}^{\otimes 2}$ decomposes as $\mathbf{1}\oplus \mathbf{3}$ but not as $\mathbf{2}\oplus \mathbf{2}$.

So, what is a nice place to learn these things for a physicist and for groups other than $SU(2)$? I am especially interested in this notation and not a "quantum mechanics" notation using raising and lowering operators on states.

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## marked as duplicate by sammy gerbil, Qmechanic♦Feb 26 '17 at 23:06

• I think this question would be better on Maths SE, because it has no physics content. – sammy gerbil Feb 26 '17 at 22:41
• I want physicists yo answer though not mathematicians. And it is directly related to physics, actually it is of huge importance to physics. – Gorbz Feb 26 '17 at 22:42
• Possible duplicate of Comprehensive book on group theory for physicists? – sammy gerbil Feb 26 '17 at 22:48
• No since my question is very specific. Decomposing a representation $SU(N)$ into sub-representations. I dont want general reference for group theory. I want to see and understand how physicists think of reps of $SU(N)$. – Gorbz Feb 26 '17 at 22:51
• @Gorbz Are you familiar with the basic of Lie algebras and their representations (concepts like weights, roots, Cartan subalgebra, step operators)? – Diracology Feb 26 '17 at 22:58

You need to learn about Clebsch-Gordan series and Young diagrams (see here) or the application of the Littlewood-Richardson rule for $su(n)$ decomposition. This is standard fare and there are plenty of textbooks around, for instance Lichtenberg's Unitary Symmetry and Elemenatry particles. There is a review by Dick Slansky that contains lots of useful stuff, but not so much on decomposing irreps. Have fun.

Edit:

The "game" can be applied in modified form to the representations of the other classical series, but the rules loose their simplicity.

If you are looking for perfectly general algorithms that will work for the classical series, then you need to look at Schur functions but this is not so easy. Basically Schur functions are characters, and one decomposes a product by expanding in Schur functions the product of two Schur functions. This is not practical to do by hand for anything but the simplest cases. Bryan Wybourne did a lot to introduce Schur functions in physics. There is also a lot of research-grade work by Ron King and his students.

If you are primarily interested in the computational aspect, there are now packages, such as LieART that work quite well. LieArt is by no means the only one and I am not endorsing this one over others; it is pretty easy to read the documentation. See also this and this from mathoverflow if you need other suggestions.

• Thanks. I have learnt once the Clebsch-Gordan series for the quantum mechanics course. Does this game apply for any compact Lie algebra? – Gorbz Feb 26 '17 at 23:30
• @Gorbz ... added some additional comments. – ZeroTheHero Feb 27 '17 at 0:22
• I think I understand how to decompose a representation in terms of Young tableaux. What I do not know is if this way works for any group though. – Gorbz Feb 28 '17 at 0:26