This post is an exact copy of one that I posted in Math's site. I do this copy because people there suggested me to do it since, apparentely, in Mathematics and Physics we use different conventions for this kind of topics and none there could understand mine. (You can see first post here: https://math.stackexchange.com/q/3134077/)
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?
My source is: http://www.th.physik.uni-bonn.de/nilles/people/luedeling/grouptheory/data/grouptheorynotes.pdf sections 6.6.1 and 6.11
No comments:
Post a Comment