# Tensor product of Spinorial Left/Right Lorentz Representation

I searched in various textbook (Zee Group and QFT, Weinberg 1-3, Cornwell, Maggiore) and all the similar question on this site but I didn’t find a final answer.

My question is simple:

• What is the decomposition of the tensor product of two Spinorial Representation $$(a,b) \otimes (c,d)$$ of the Lorentz Group?

I know that $$(a,b) \otimes (c,d)= (a\otimes c, b \otimes d)$$, and I also know how to compose angular momentum, so I am trying to find how to express $$(a\otimes c, b \otimes d)$$ as a direct sum of different terms.

Since we have 2 tensor product, I would naively say that:

$$(a\otimes c, b \otimes d) = (a+c,b+d) \oplus (a+c-1,b+d) \oplus \dots \oplus (|a-c|,b+d) \oplus (a+c,b+d-1) \oplus \dots \oplus (|a-c|,b+d-1) \oplus (a+c,|b-d|) \oplus \dots \oplus (|a-c|, |b-d|)$$

But since

$$(\frac{1}{2},\frac{1}{2})\otimes \left((0,\frac{1}{2}) \oplus (\frac{1}{2},0)\right)=(1, \frac{1}{2} )\oplus (\frac{1}{2},0)\oplus( \frac{1}{2},1) \oplus(0, \frac{1}{2})$$

My formula seems wrong.