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I am confused about the notation for representations of $\mathfrak{su}(3)$. Often a bold number is used to denote a particular representation e.g. $\mathbf{3}$ is used to denote the fundamental representation and $\mathbf{8}$ denotes the adjoint representation. Presumably, this notation is only justified if there is a unique representation of a given dimension, $n$, and then we can denote this representation by $\mathbf{n}$. I don't know how to prove this if it is true. If it is not true, then what is meant by the representation $\mathbf{n}$?

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  • $\begingroup$ I have seen an answer to this before, but, my searching powers are insufficient to locate it... however, what about $\bar{3}$ ? As I recall, both types are used to create the representations of interest... $\endgroup$ Jul 26, 2014 at 0:56
  • $\begingroup$ Thanks for your reply. I couldn't find anything when I searched either. $\mathbf{\bar{3}}$ is another representation of $\mathfrak{su}(3)$. Is it equivalent? If there are several inequivalent representations of some dimension $n$, it doesn't seem that $\mathbf{n}$ is well-defined, unless $\mathbf{3}$, $\mathbf{8}$ etc. are the only representations which have this bold number notation and are just defined by convention. $\endgroup$
    – octopus
    Jul 26, 2014 at 1:03
  • $\begingroup$ Well, I gave a talk a few years ago see supermath.info/quarks.pdf see page 13 in particular, the $3$ and $\bar{3}$ are not equivalent. That said, I'm not well-versed in the proper mathematics of the representation theory to give a satisfactory answer here. There are a number of users who are and this will get answered soon I suspect... $\endgroup$ Jul 26, 2014 at 2:31

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It is not true that a (finite-dimensional, complex) representation of $\mathfrak{su}(3)$ is determined by its dimension. In fact it's already the case, as James Cook points out in the comments, that the defining representation $V$ (which I think is the representation you denote by $\mathbf{3}$) and its dual $V^{\ast}$ are not isomorphic. One way to see this is that the corresponding representations of $\text{SU}(3)$ are not isomorphic because they have different characters.

So I think this is a notational convention as you suspect in the comments. It is true that representations of $\mathfrak{su}(2)$ are determined by their dimension, so maybe that's where the convention comes from. Incidentally, the representation theory of all of these classical Lie algebras is completely understood; pop open any textbook on the subject to find the general statements.

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  • $\begingroup$ How do you write the representation $3$ without a computer : do we write just "3", or with a vector on top of it ? As $\vec{3}$ ? $\endgroup$ Jun 6, 2020 at 17:40

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