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For each finite irreducible representation of Lie algebra $su(3)$ one knows that it is characterized by highest weight $(\lambda_1, \lambda_2)$ with integral entries. In this notation, $(1,0)$ is fundamental representation, $(0,1)$ is anti-fundamental representation, and $(1,1)$ is adjoint representation.

Given three highest weight vectors, say $(\lambda_1, \lambda_2)$ and $(\mu_1, \mu_2)$, $(\nu_1,\nu_2)$, is there an arithmetic way to determine if $(\nu_1,\nu_2)$ is in the tensor product decomposition of $(\lambda_1, \lambda_2) \otimes (\mu_1, \mu_2)$?

The analogy of this is the tensor product decompostion of $su(2)$, where we know $j$ is in $j_1 \otimes j_2$ if $$|j_1 - j_2| \leq j\leq j_1+j_2.$$ Is the same thing happen for $su(3)$ so that one does not need to go through Young tableau prescription?

Qmechanic
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Kevin Ye
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  • Even in the case of $\mathfrak{su}(2)$ you have a parity condition in addition to the listed inequalities: $j$ needs to share the same parity as $j_1+j_2$. When $m\ge n$ the tensor product decomposes like $$V(m)\otimes V(n)=V(m+n)\oplus V(m+n-2)\oplus V(m+n-4)\oplus\cdots\oplus V(m-n).$$ In the case of $\mathfrak{su}(3)$ the root lattice has index three inside the weight lattice, resulting in a modulo three condition instead of a parity test. – Jyrki Lahtonen Jan 28 '25 at 14:06
  • (cont'd) Anyway, with $\mathfrak{su}(3)$ I usually calculate these using the formula explained here. It is explained in Humphreys's book. Easier to explain, when you have pictures of weight lattices at hand :-) – Jyrki Lahtonen Jan 28 '25 at 14:09

1 Answers1

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  • The $su(2)$ fusion rule is $$ \begin{align} (n)\otimes (m) ~\cong~& \bigoplus_{\ell\in\mathbb{N}_0} \underbrace{N^{\ell}_{n,m}}_{\in\mathbb{N}_0} (\ell) \cr ~\cong~& \bigoplus_{a=0}^{\min(n,m)}(n+m-2a) \end{align}$$ for $n,m\in\mathbb{N}_0$, where $$ N^{\ell}_{n,m} ~=~\left\{\begin{array}{rl}1&\text{if}\quad |n-m|\leq\ell\leq n+m\quad\wedge\quad n+m-\ell\in 2\mathbb{Z},\cr 0&\text{otherwise}.\end{array} \right.$$ (The above integers are 2 times what physicists call spin.)

  • The $su(3)$ fusion rule is $$\begin{align} (n,\bar{n})&\otimes (m,\bar{m})\cr ~\cong~& \bigoplus_{(\ell,\bar{\ell})\in\mathbb{N}_0^2} \underbrace{N^{(\ell,\bar{\ell})}_{ (n,\bar{n})(m,\bar{m})}}_{\in\mathbb{N}_0} (\ell,\bar{\ell})\cr ~\cong~& \bigoplus_{k=0}^{\min(n,\bar{m})}\bigoplus_{\bar{k}=0}^{\min(\bar{n},m)}\left\{\bigoplus_{a=0}^{\min(n-k,m-\bar{k})} (n+m-k-\bar{k}-2a,~\bar{n}+\bar{m}-k-\bar{k}+a)\right.\cr ~\oplus~& \left.\bigoplus_{\bar{a}=1}^{\min(\bar{n}-\bar{k},\bar{m}-k)}(n+m-k-\bar{k}+\bar{a},~\bar{n}+\bar{m}-k-\bar{k}-2\bar{a})\right\} \end{align} $$ for $n,\bar{n},m,\bar{m}\in\mathbb{N}_0$, as explained in my Phys.SE answer here.

    Necessary (but not sufficient) selection rules:

    • $ \ell,\bar{\ell}~\in~\mathbb{N}_0, $
    • $ (n+m-\ell) +2(\bar{n}+\bar{m}-\bar{\ell})~\geq~0,$
    • $ 2(n+m-\ell) +(\bar{n}+\bar{m}-\bar{\ell})~\geq~0,$
    • $ (n-\bar{n})+(m-\bar{m})-(\ell-\bar{\ell})~\in~3\mathbb{Z}.$

    enter image description here

    $\uparrow$ Fig. 1. Schematic illustration of possible $(\ell,\bar{\ell})$ weights of irreps in an $(n,\bar{n})\otimes (m,\bar{m})$ tensor representation of $su(3)$. NB: Multiplicities are not indicated.

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