# Finding $\mathbf{10}\otimes \mathbf{8}\otimes \mathbf{8}\otimes \mathbf{8}$ in $SU(3)$

I know that in $SU(3)$ $$\mathbf{8}\otimes \mathbf{8} = \mathbf{27}+\mathbf{10}+\mathbf{\bar{10}}+\mathbf{8}+\mathbf{8}+\mathbf{1}.$$

How can one use this to compute $$\mathbf{10}\otimes \mathbf{8}\otimes \mathbf{8}\otimes \mathbf{8}?$$

$$\tag{1} \mathbf{10}\otimes \mathbf{8}\otimes \mathbf{8} = 10\otimes27 \\ +10\otimes10 \\+10\otimes\bar{10} \\+10\otimes8 \\+10\otimes8 \\+10\otimes1?$$

Is $(1)$ even ok?

• Could you give a little context, or at least definitions? To the best of my knowledge, 10 and 8 are integers, not three-by-three unitary matrices.
– Neal
Jan 24, 2015 at 0:57
• @Neal see the quark model Jan 24, 2015 at 1:21
• Yes, tensor products of reps distribute over sums of reps, so (1) is ok. I'm not sure how, e.g. $10\otimes 27$ decomposes. Jan 24, 2015 at 1:50
• Can one reduce 10x1? I strongly suspect no,but I have to ask. Jan 24, 2015 at 1:56
• @Omnomnomnom Thanks!
– Neal
Jan 24, 2015 at 3:04

Here is a portion of the answer. All of these calculations are coming from here.

In terms of that website, we think of $SU(3)$ as $A_2$. Then, the $10$ dim rep has heighest weight $(3,0)$ in their notation (and the $\overline{10}$ has heighest weight $(0,3)$). The $27$ d rep is $(2,2)$. The notation the website uses is $X[3,0]$, $X[0,3]$, or $X[2,2]$ respectively.

According to that website, we have

$$10\otimes 27 = 1X[5,2] +1X[3,3] +1X[4,1] +1X[1,4] +1X[2,2] +1X[3,0] +1X[0,3] +1X[1,1],$$ or in your notation,

$$10\otimes 27 = 81 + 64 + 35 + \overline{35} + 27 + 10 + \overline{10} + 8$$

A small disclaimer: in $SU(3)$, the dimension of a representation does not determine the representation, even discounting conjugate representations. For example, $X[2,1]$, $X[4,0]$, $X[1,2]$, and $X[0,4]$ are each distinct irreducible $15$ dimensional representations. So, notation like $8$, $10$, $\overline{10}$, while unambiguous, would be ambiguous applied to the number $15$.

• To answer your comment, yes, $10\otimes 1 = 10$. Also, I'm not sure what size inputs the program I linked to can handle (I think it can only handle single digit highest weights). Jan 24, 2015 at 2:10
• Great answer @Jason DeVito. Where did you find the dictionary for that website's notation? Jan 26, 2015 at 15:06
• @LoveLearning: I wish I had a nice answer. I used the website quite a bit about 6-7 years ago when trying to figure out some stuff for my thesis, and I remember a lot of how it worked from then. The $A_2 \iff SU(3)$ notation is fairly standard (see classification of simple Lie algebras), and the recognition that, e.g., $X[3,0]\iff 10$ came from plugging things into their "dimension of a module" option, and noting that dimension $10$ only occurred for $X[3,0]$ and $X[0,3]$. Jan 26, 2015 at 16:26
• Well, I kind of cheated. I don't know your notation well enough to say which one is barred, so in my head, I defined $X[4,1] = 35$. In terms of the Dynkin indices, it then follows that $X[1,4] = \overline{35}$. (I'm actually heading to lunch, then class, so I won't be able to respond for another 4 hours or so...) Jan 26, 2015 at 16:45
• Not to my knowledge. The only notation I've ever used is by labeling the nodes in the Dynkin diagram by natural numbers, or variations on that. Jan 27, 2015 at 14:09