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I wasn't sure how to write the title because I don't really understand this topic. Here's my question: When we are constructing hadrons we put quarks together to form higher representations of the flavour symmetry $SU(3)$ (which I know is not exact, but let's deal only with three quarks for now). Mesons are built with a quark and an antiquark

\begin{equation} 3\otimes \bar{3}=8\oplus1 \end{equation}

This equation can actually be written in a tensorial way like this

\begin{equation} q_i\bar{q}_j=\big(q_i\bar{q}_j-\frac{1}{3}q_k\bar{q}_k\delta_{ij}\big)+\frac{1}{3}q_k\bar{q}_k\delta_{ij} \end{equation}

Since the last term is a singlet, the first term must be the octet. This is super useful because now we know how to write the octet and the singlet representations in terms of the foundamental quarks. I was wondering how to do this with the baryons. They are built with three quarks

\begin{equation} 3\otimes 3\otimes3=10\oplus8\oplus8\oplus1 \end{equation}

but I don't know how to write this in a tensorial way like with the mesons. I looked all over the literature but I only found the meson formula. How is the decomposition of three quarks writen in a tensorial way?

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  • $\begingroup$ Possible duplicate: physics.stackexchange.com/q/167680/2451 $\endgroup$
    – Qmechanic ♦
    Commented Mar 3, 2019 at 2:42
  • $\begingroup$ Related : Symmetry in terms of matrices $\endgroup$
    – Voulkos
    Commented Mar 3, 2019 at 3:09
  • $\begingroup$ I understand how those links (and the links quoted there) are related to my question. However, they don't give an answer in terms of a tensorial formula, or if they do, they use tools I really don't know how to handle, like young tableau. I would like a decomposition in terms of tensorial objects. $\endgroup$
    – P. C. Spaniel
    Commented Mar 3, 2019 at 4:51

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