# Simplifying a direct sum $\mathbf{3}\oplus\mathbf{3}\oplus\mathbf{2}$ etc

In particle physics, one often uses the dimensionality of the irrep to label the irrep (apparently this is not a very good idea since the dimension does not unambiguously determine the rep.). What are the rules one can use to simplify calculations involving direct sums. I have this one example of a direct product that on the right hand side contains many repeating terms:

For instance, suppose you have something like $D = 10\otimes 8 \oplus 10\otimes 8.$ If I have done my calculations right one should get for each term in $SU(3)$

$$\mathbf{10}\otimes \mathbf{8} = \mathbf{35}+\mathbf{27}+\mathbf{10}+\mathbf{8}$$ so that (dropping the bold notation) $$\tag{1}10\otimes 8\oplus 10\otimes 8 = 35\oplus27\oplus10\oplus8\oplus 35\oplus27\oplus10\oplus8.$$

Is there a way to simplify the right hand side of equation $(1)$ in the sense of simply writing e.g. $$\mathbf{35}\oplus\mathbf{35}\oplus\mathbf{35} = 3\cdot \mathbf{35}$$ where $3\in \mathbb{Z}.$

• How do you define directs sums / tensor products of numbers? – Marc van Leeuwen Jan 26 '15 at 15:53
• I've updated the answer @MarcvanLeeuwen – Your Majesty Jan 26 '15 at 16:01
• You can simplify it just as you did, but you have to be very careful that you keep bold/nonbold stuff straight. Also, your original calculation could have simplified to $2(\mathbf{10}\otimes \mathbf{8})$. – Jason DeVito Jan 26 '15 at 21:14
• Yeah I suspected that. – Your Majesty Jan 26 '15 at 22:09