14d. Clebsch-Gordan rules

[math] \newcommand{\mathds}{\mathbb}[/math]

As a last piece of pure algebra, we are now in position of dealing, in a quite conceptual way, with [math]SU_2[/math] and [math]SO_3[/math]. Regarding [math]SU_2[/math], the result here is as follows:

Theorem

The irreducible representations of [math]SU_2[/math] are all self-adjoint, and can be labelled by positive integers, with their fusion rules being as follows,

[[math]] r_k\otimes r_l=r_{|k-l|}+r_{|k-l|+2}+\ldots+r_{k+l} [[/math]]

called Clebsch-Gordan rules. The corresponding dimensions are [math]\dim r_k=k+1[/math].

Show Proof

There are several proofs for this fact, the simplest one, with the knowledge that we have, being via purely algebraic methods, as follows:

(1) Our first claim is that we have the following estimate, telling us that the even moments of the main character are smaller than the Catalan numbers:

[[math]] \int_{SU_2}\chi^{2k}\leq C_k [[/math]]

But this is something that we know from chapter 12, obtained by using [math]SU_2\simeq S^3_\mathbb R[/math] and spherical integrals, and with the stronger statement that we have in fact equality [math]=[/math]. However, for the purposes of what follows, the above [math]\leq[/math] estimate will do.

(2) Alternatively, the above estimate can be deduced with purely algebraic methods, by using an easiness type argument for [math]SU_2[/math], as follows:

[[math]] \begin{eqnarray*} \int_{SU_2}\chi^{2k} &=&\dim(Fix(u^{\otimes 2k}))\\ &=&\dim\left(span\left(T_\pi'\Big|\pi\in NC_2(2k)\right)\right)\\ &\leq&|NC_2(2k)|\\ &=&C_k \end{eqnarray*} [[/math]]

To be more precise, [math]SU_2[/math] is not exactly easy, but rather “super-easy”, coming from a different implementation [math]\pi\to T_\pi'[/math] of the pairings, involving some signs. And with this being proved exactly as the Brauer theorem for [math]O_N[/math], with modifications where needed.

(3) Long story short, we have our estimate in (1), and this is all that we need. Our claim is that we can construct, by recurrence on [math]k\in\mathbb N[/math], a sequence [math]r_k[/math] of irreducible, self-adjoint and distinct representations of [math]SU_2[/math], satisfying:

[[math]] r_0=1\quad,\quad r_1=u\quad,\quad r_k+r_{k-2}=r_{k-1}\otimes r_1 [[/math]]

Indeed, assume that [math]r_0,\ldots,r_{k-1}[/math] are constructed, and let us construct [math]r_k[/math]. We have:

[[math]] r_{k-1}+r_{k-3}=r_{k-2}\otimes r_1 [[/math]]

Thus [math]r_{k-1}\subset r_{k-2}\otimes r_1[/math], and since [math]r_{k-2}[/math] is irreducible, by Frobenius we have:

[[math]] r_{k-2}\subset r_{k-1}\otimes r_1 [[/math]]

We conclude there exists a certain representation [math]r_k[/math] such that:

[[math]] r_k+r_{k-2}=r_{k-1}\otimes r_1 [[/math]]

(4) By recurrence, [math]r_k[/math] is self-adjoint. Now observe that according to our recurrence formula, we can split [math]u^{\otimes k}[/math] as a sum of the following type, with positive coefficients:

[[math]] u^{\otimes k}=c_kr_k+c_{k-2}r_{k-2}+\ldots [[/math]]

We conclude by Peter-Weyl that we have an inequality as follows, with equality precisely when [math]r_k[/math] is irreducible, and non-equivalent to the other summands [math]r_i[/math]:

[[math]] \sum_ic_i^2\leq\dim(End(u^{\otimes k})) [[/math]]

(5) But by (1) the number on the right is [math]\leq C_k[/math], and some straightforward combinatorics, based on the fusion rules, shows that the number on the left is [math]C_k[/math] as well:

[[math]] C_k=\sum_ic_i^2\leq\dim(End(u^{\otimes k}))=\int_{SU_2}\chi^{2k}\leq C_k [[/math]]

Thus we have equality in our estimate, so our representation [math]r_k[/math] is irreducible, and non-equivalent to [math]r_{k-2},r_{k-4},\ldots[/math] Moreover, this representation [math]r_k[/math] is not equivalent to [math]r_{k-1},r_{k-3},\ldots[/math] either, with this coming from [math]r_p\subset u^{\otimes p}[/math] for any [math]p[/math], and from:

[[math]] \dim(Fix(u^{\otimes 2s+1}))=\int_{SU_2}\chi^{2s+1}=0 [[/math]]

(6) Thus, we proved our claim. Now since each irreducible representation of [math]SU_2[/math] appears into some [math]u^{\otimes k}[/math], and we know how to decompose each [math]u^{\otimes k}[/math] into sums of representations [math]r_k[/math], these representations [math]r_k[/math] are all the irreducible representations of [math]SU_2[/math], and we are done with the main assertion. As for the dimension formula, this is clear.

■

Regarding now [math]SO_3[/math], we have here a similar result, as follows:

Theorem

The irreducible representations of [math]SO_3[/math] are all self-adjoint, and can be labelled by positive integers, with their fusion rules being as follows,

[[math]] r_k\otimes r_l=r_{|k-l|}+r_{|k-l|+1}+\ldots+r_{k+l} [[/math]]

also called Clebsch-Gordan rules. The corresponding dimensions are [math]\dim r_k=2k+1[/math].

Show Proof

As before with [math]SU_2[/math], there are many possible proofs here, which are all instructive. Here is our take on the subject, in the spirit of our proof for [math]SU_2[/math]:

(1) Our first claim is that we have the following formula, telling us that the moments of the main character equal the Catalan numbers:

[[math]] \int_{SO_3}\chi^k=C_k [[/math]]

But this is something that we know from chapter 12, coming from Euler-Rodrigues. Alternatively, this can be deduced as well from Tannakian duality, a bit as for [math]SU_2[/math].

(2) Our claim now is that we can construct, by recurrence on [math]k\in\mathbb N[/math], a sequence [math]r_k[/math] of irreducible, self-adjoint and distinct representations of [math]SO_3[/math], satisfying:

[[math]] r_0=1\quad,\quad r_1=u-1\quad,\quad r_k+r_{k-1}+r_{k-2}=r_{k-1}\otimes r_1 [[/math]]

Indeed, assume that [math]r_0,\ldots,r_{k-1}[/math] are constructed, and let us construct [math]r_k[/math]. The Frobenius trick from the proof for [math]SU_2[/math] will no longer work, due to some technical reasons, so we have to invoke (1). To be more precise, by integrating characters we obtain:

[[math]] r_{k-1},r_{k-2}\subset r_{k-1}\otimes r_1 [[/math]]

Thus there exists a representation [math]r_k[/math] such that:

[[math]] r_{k-1}\otimes r_1=r_k+r_{k-1}+r_{k-2} [[/math]]

(3) Once again by integrating characters, we conclude that [math]r_k[/math] is irreducible, and non-equivalent to [math]r_1,\ldots,r_{k-1}[/math], and this proves our claim. Also, since any irreducible representation of [math]SO_3[/math] must appear in some tensor power of [math]u[/math], and we can decompose each [math]u^{\otimes k}[/math] into sums of representations [math]r_p[/math], we conclude that these representations [math]r_p[/math] are all the irreducible representations of [math]SO_3[/math]. Finally, the dimension formula is clear.

■

There are of course many other things that can be said about [math]SU_2[/math] and [math]SO_3[/math]. For instance, with the proof of Theorem 14.25 and Theorem 14.26 done in a purely algebraic fashion, by using the super-easiness property of [math]SU_2[/math] and [math]SO_3[/math], the Euler-Rodrigues formula can be deduced afterwards from this, without any single computation, the argument being that by Peter-Weyl the embedding [math]PU_2\subset SO_3[/math] must be indeed an equality.

As a conclusion to all this, you have now a decent level in group theory, and algebra in general, and you can start if you want exploring all sorts of other things, such as:

(1) Quantum groups. This is something modern and interesting, inspired by quantum mechanics, and the algebra is not that much complicated than what we did in the above. The must-read papers here are the one of Drinfeld on one hand, and the one of Woronowicz on the other. As for books, you have Chari-Pressley for the Drinfeld quantum groups, and my book for Woronowicz quantum groups.

(2) Planar algebras. This is something very related to the quantum groups, and perhaps even more exciting than them, due to all sorts of pictures, and relations with modern physics, developed by Jones in , , , . With all sorts of interesting ramifications, and you can check here too the classical books or papers of Atiyah , Di Francesco , Temperley-Lieb , Witten and Zwiebach .

But since we are now towards the end of the present book, better stay with us, and you can look into all this afterwards. We still have all sorts of interesting things to be done, including applying all the algebra that we learned, to questions in probability.

General references

Banica, Teo (2024). "Linear algebra and group theory". arXiv:2206.09283 [math.CO].