# Clebsch–Gordan decomposition formula for algebraic groups

$$\DeclareMathOperator\SL{SL}$$There is a well-known Clebsch–Gordan decomposition formula for irreducible representations of $$\SL_2$$. If $$V_n$$ denotes the unique $$n+1$$-dimensional irreducible representation of $$\SL_2$$ we have that for $$n \geq m$$:

$$V_n \otimes V_m = V_{n+m} \oplus V_{n+m-2} \oplus \dots \oplus V_{n-m}.$$

I wonder if similar formulas exist for any other algebraic groups, for instance, $$\SL_3$$ or the symmetric group $$S_n$$.

Böhning and Bothmer - A Clebsch–Gordan formula for $$\SL_3(\mathbb C)$$ and applications to rationality uses an algorithm with Young tableaux to decompose the tensor products of two irreducible representations of $$\SL_3$$. More about this algorithm can be found in chapter 12 of Georgi - Lie algebras in particle physics. For $$S_n$$, I found Schindler - The decomposition of the tensor product of representations of the symmetric group, but no explicit formula is given.

• You should read Fulton and Harris' book on representations of Lie algebras, where they give a wealth of information about decomposition of tensor products of irreducible representations ("plethysm", as they call it). Jan 27 at 11:38
• I already had a look at chapter 13 of Fulton and Harris. From exercise $13.8$ it's easy to deduce a formula for the decomposition into irreducibles of $V \otimes \Gamma_{a,b}$, where $V$ is the standard representation and $\Gamma_{a,b}$ irreducible. However, I would be more interested in finding a formula for $\Gamma_{a,b} \otimes \Gamma_{c,d}$. Jan 27 at 11:59
• You will certainly find something useful in Table 5 of this book: A. L. Onishchik, E. B. Vinberg, Lie groups and algebraic groups. Springer-Verlag, Berlin, 1990. Jan 27 at 13:21
• I have been looking at table 5 of "Lie groups and algebraic groups". They say that $R(\pi_p) R(\pi_q) = \sum_{i \geq 0} R(\pi_{p+i} + \pi_{q-i})$, but I am not sure what $R(\pi_q)$ means as they only define $R(\pi_1)$ as the simplest representation. I have looked around other sections of the book, but I can't find where they introduce this notation. Jan 27 at 14:59
• In the table for $A_l$ they write $\bigwedge^p R=R(\pi_p)$. Therefore, I think that $\pi_p$ is the weight with numerical labels $\langle \pi_p,\alpha_i^\vee\rangle =\delta_{p,i}\,$, where $\alpha_i^\vee$ are the simple coroots, and $\delta_{p,i}$ denotes Kronecker's symbol. Jan 28 at 9:42

Let $$G$$ denote the group, and suppose one has an enumeration of its irreducible representations $$V_{\lambda}$$ by some combinatorial objects $$\lambda$$, like integers for $$SU_2$$ (or finite dimensional non-unitary representations of $$SL_2$$) or integer partitions, etc. There are two different Clebsch-Gordan (CG) problems which the OP seems to conflate.

CG1) The numerical CG problem: It is to figure out the multiplicities $$m(\lambda,\mu;\nu)\in\mathbb{N}$$ in the general decomposition into irreducibles of tensor products of two irreducibles: $$V_{\lambda}\otimes V_{\mu}=\bigoplus_{\nu}V_{\nu}^{\oplus m(\lambda,\mu;\nu)}\ .$$

CG2) The explicit CG problem: It is to realize the above decomposition with explicit intertwiners, namely, to write a decomposition of the identity operator $$I_{V_{\lambda}\otimes V_{\mu}}$$ on $$V_{\lambda}\otimes V_{\mu}$$ in the form: $$I_{V_{\lambda}\otimes V_{\mu}}=\sum_{\nu}\sum_{j=1}^{m(\lambda,\mu;\nu)} \iota_{\lambda,\mu,\nu,j}\circ\pi_{\lambda,\mu,\nu,j}$$ where $$\pi_{\lambda,\mu,\nu,j}\in {\rm Hom}_G(V_{\lambda}\otimes V_{\mu},V_{\nu})$$ and $$\iota_{\lambda,\mu,\nu,j}\in {\rm Hom}_G(V_{\nu},V_{\lambda}\otimes V_{\mu})$$ are explicit $$G$$-equivariant maps, i.e., intertwiners.

Note that to be able to even ask the question, a prerequisite is to solve

CG0) The parametrization of irreducibles: Namely, understanding the list of irreducibles, and having a parametrization $$\lambda\mapsto V_{\lambda}$$.

For $$SU_2$$, $$SL_2$$ all these problems were solved by Paul Gordan and Alfred Clebsch in the mid 19-th century, see Section 2 of my article:

Problem CG1 for $$SU_n$$, $$SL_n$$, $$GL_n$$ has been solved, and the multiplicities are the so-called Littlewood-Richardson coefficients. For $$S_n$$, CG1 is much more difficult. The multiplicities are the Kronecker coefficients, and there is no satisfactory combinatorial description for them.

The recent article by Böhning and Graf von Bothmer does not just solve CG1 for $$SU_3$$, $$SL_3$$ (that's known from a long time ago), but rather problem CG2 for these groups. The case of $$SL_n$$, $$GL_n$$ is still open. When $$\mu$$ is the fundamental representation (adding a single box), there are some results, see

• M. Hunziker, J. A. Miller, and M. Sepanski. Explicit Pieri Inclusions. Electronic J. of Combinatorics 28 (2021), no. 3, P3.49.

and references therein (in particular some older work by Peter Olver). As for CG0 in the case or $$SL_n$$, it was solved by Alfred Clebsch in the 1870's, and later by Deruyts, and then Schur, see:

For $$S_n$$, CG0 was solved by Alfred Young and later by Specht. A good account is in the lectures by Adriano Grasia Alfred Young’s construction of the irreducible representations of $$S_n$$.

Finally, the only instance of CG2 related work for $$S_n$$ that I am aware of is:

• From memory, it is not done in Fulton-Harris but it is done in the book by Fulton alone on Young Tableaux. Jan 27 at 16:28
• For algorithms for the computation of LR coefficients the wikipedia page is not bad as a starting point en.wikipedia.org/wiki/Littlewood%E2%80%93Richardson_rule and also a page by Per Alexandersson symmetricfunctions.com/… for quickest ways to compute see the paragraph LR tableaux in Per's page. Jan 27 at 16:36
• The Littlewood-Richardson rule is discussed in detail in Appendix 1 of Chapter 7 (written by Sergey Fomin) in Stanley's "Enumerative Combinatorics," Vol. 2, and the connection with general linear group characters is explained in Appendix 2. Jan 27 at 16:40
• For one-off calculations within CG1, you can also refer to the LiE computer algebra package. It can handle any of the complex simple Lie algebras. The biggest hassle of the UI, depending on your background, is learning to label irreps by their highest weight vectors. Jan 27 at 21:53
• Okay, this looks like a reasonable survey for semisimple algebraic groups $G$: "Tensor product decomposition" by Shrawan Kumar, in the 2010 ICM Proceedings, available online at kumar.math.unc.edu/papers/kumar60.pdf Jan 28 at 18:09