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I've been trying to figure out this statement from the PDG quark model summary (PDF).

Following $\mathrm{SU}(3)$, the nine possible $q\bar{q}′$ combinations containing the light $u$, $d$, and $s$ quarks are grouped into an octet and a singlet of light quark mesons:

$\mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{8} \oplus \mathbf{1}$

This looks like some very nice notation that comes up now and then, but unfortunately I don't have any idea what it's called and the introductory group theory material I've skimmed doesn't explain it. In particular I'm trying to figure out what the $\otimes$ and $\oplus$ mean.

Obviously this is a simple question to answer with Google, but that's a bit hard without knowing what this is called. What do I have to google to find out more about this?

Shep
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1 Answers1

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In physicist jargon, we talk about group representations of $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$ by denoting an irreducible representation whose representation vector space has dimension $N$ by $\mathbf{N}$.

Hence, the statement $\mathbf{3} \otimes \bar{\mathbf{3}} = \mathbf{1} \oplus \mathbf{8}$ is the statement that the tensor product of the three-dimensional representation of $\mathrm{SU}(3)$ (also called its fundamental representation, as it is the smallest non-trivial one) and its conjugate representation decomposes as the direct sum of the eight-dimensional (the adjoint representation) and the trivial representation.

The notation works for $\mathrm{SU}(3)$ because there are only two irreducible representations with a given dimension, and they are conjugates of each other, so $\mathbf{N}$ and $\bar{\mathbf{N}}$ are sufficient to denote all possible (finite-dimensional) representations

ACuriousMind
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