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?

  • $\begingroup$ wikipedia doesn't use this notation to explain SU(2) or group representation theory, any idea who would? $\endgroup$ – Shep Jan 16 '15 at 17:24
  • $\begingroup$ Related: physics.stackexchange.com/q/41424/2451 and links therein. $\endgroup$ – Qmechanic Jan 16 '15 at 19:07
  • $\begingroup$ @Shep A lot of graduate physics texts, and Lie group/representation theory texts. $\endgroup$ – JamalS Jan 16 '15 at 19:19
  • 4
    $\begingroup$ Something useful to keep in mind: when mathematicians say "introductory group theory" they are thinking of stuff like the Sylow theorems. When physicists say "group theory" they really mean representation theory or module theory or linear algebra -- basically most of introductory abstract algebra except group theory proper. $\endgroup$ – user10851 Jan 16 '15 at 19:21
  • $\begingroup$ @Shep So, I had a look around, and Wikipedia does use this notation, see for example en.wikipedia.org/wiki/Grand_Unified_Theory, in the $SU(5)$ section. $\endgroup$ – JamalS Jan 18 '15 at 18:51

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


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