# Complex conjugated representation and Young tableaux

Imagine you have the Young tableu and the Dynkin numbers, $$(q_1, q_2, ..., q_r)$$, of the Lie algebra of $$SU(n)$$ which has $$r$$ simple roots. The way I assign Dynkin numbers is increasing its value from left to right so the $$k$$-th Dynkin number is the number of columns with $$k$$ boxes: $$q_k$$ columns made of $$k$$ boxes.

The Young tableau is the 'usual' one with columns that decreases in boxes from left to right. The calculation of the dimension gives you some number $$d$$ that is given by

$$d = \frac{N}{H}$$

Where $$N$$ is the product of the following numbers: in the highest left box for $$SU(n)$$ write an $$n$$ and going to the right, increase this number in one unit box per box. Going down, decrease the number in the same amount box per box. $$N$$ is the product of all those numbers. $$H$$ is the product of the hook numbers: in each box write the number of boxes that you cut going from right (out of tableaux) to left till you reach that box and then keep cutting boxes going down from that box. Do this for each box and the product of these numbers (hook numbers) is $$H$$.

Now, my question is: how can I know if this Young tableau corresponds to the representation $$d$$ or to the complex conjugated $$\bar{d}$$ since both of them have the same dimension?

• Let me guess: physicist? – Matt Samuel Mar 3 at 22:23
• Hahahaha... busted! But how is that helping me? – Vicky Mar 3 at 22:27
• I'd help if I could, but I don't know the physicists' conventions for matching the representation to its name. It doesn't make sense to me, because there can be representations of the same dimension that are not conjugates of one another. – Matt Samuel Mar 3 at 22:29
• For $SU(n)$ you can find 2 Young tableaux with the same dimension: one for the complex conjugated and one for the non-conjugated. My question is: how do I know which tableau matches which representation? I think we do not have any convention because I've nerver heard of them – Vicky Mar 3 at 22:34
• You need to provide more context. What kind of Young tableau are you allowing? Is $r=n-1$? What are the rules for assigning Dynkin numbers and how is this related to the Young tableau? How do you calculate the dimension? Can you describe how a diagonal matrix acts? – David Hill Mar 7 at 17:01