# Tensor methods for representations of SU(n)

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?

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