More than one 15 dimensional irrep of $\mathfrak{su}(3)$

Question

The irreps of (complexified) $\mathfrak{su}(n)$ are labelled by highest weight Dynkin labels $(a_1, \ldots, a_{n-1})$, and are often referred to simply by their dimension, e.g. $\mathbf{3}$ to label $(1, 0)$ of $\mathfrak{su}(3)$. Now consider the $(2, 1)$ and $(4, 0)$ irreps of $\mathfrak{su}(3)$: both have dimension 15, but they are not dual (the dual of $(2, 1)$ is $(1, 2)$). Is there a standard way in the physicists notation to distinguish such irreps? If not, then is there ever any (physical) need to do so?

Answer 1

? Of course physicists use the standard Dynkin indices, as you confirm by using your Slansky 1981, Table 23, p 92.

Even though your two irreps have the same $$ d(p,q)= \tfrac{1}{2} (p+1)(q+1)(p+q+2) ~~~~\leadsto ~~~ 15, $$ Their quadratic Casimirs entering in the β function of QCD are different, eigenvalues $$ (p^2+q^2+3p+3q+pq)/3 ~~~~\leadsto ~~~ 28/3 \leftrightarrow 16/3, $$ respectively; and the cubic Casimir eigenvalues, rescaled to the respective anomaly coefficients, are $$ (p-q)(3+p+2q)(3+q+2p)/18 ~~~~\leadsto ~~~ 154/9 \leftrightarrow 28/9 , $$ so they'd contribute differently to anomalies.

Answer 2

The irreps are really different, as illustrated by branching rules to subgroup and other relevant quantum numbers.

For instance the $(4,0)$ contains angular momenta $L=4,2,0$ but the $(2,1)$ contains $L=3,2,1$. The branch to $\mathfrak{su}(2)$ irreps is also different.

The irrep $(4,0)$ does not have weight multiplicities but the irrep $(2,1)$ has three weights occurring twice. Thus the physical contents are certainly different.