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I'm familiar with the Young Tableaux method for finding irreducible representations of $\mathrm{SU}(n)$ for $n\in\mathbb{N}$, and I also know of the tensor methods for finding irreducible representations of $\mathrm{SU}(3)$.

Tensor methods for $\mathrm{SU}(3)$ seem to rely heavily on the fact that irreducible representations of this group are identified by two natural numbers $(p,q)$, which define the covariance and contravariance of the tensor associated with that representation. But, if I'm not wrong, and, $n-1$ are needed to identify every irreducible representation of $\mathrm{SU}(n)$. I have no proof of this but it makes sense if we look at the Young Tableaux method.

My questions, then, are:

  • Do tensor methods for finding irreducible representations of $\mathrm{SU}(n)$ exist for $n > 3$?

  • If so, which are those?

  • If not, how is it that they exist for $n \leq 3$, why do they work?

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  • $\begingroup$ The irreducible representations are determined by their restriction to a maximal torus which has dimension $n-1$, these are the weights. $\endgroup$ – Charlie Frohman Jan 29 at 2:27
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    $\begingroup$ Yes, these are the Dynkin indices: the number of single boxes, two-box columns, 3-box columns...etc. You want a tutorial? Slansky 1981. $\endgroup$ – Cosmas Zachos Feb 4 at 0:01

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