# Clebsch-Gordan coefficient for 1x0

I'm trying to work out the combination of $$|1\ 0 \rangle|0\ 0 \rangle$$ (in this case they represent isospin, $$|I\ I_3 \rangle$$) using Clebsch-Gordan coefficients, but the table for $$j_1\times j_2=1\times0$$ doesn't appear in the table of Clebsch-Gordan coefficients, they always start at $$\frac{1}{2} \times \frac{1}{2}$$. Does this mean this combination is not possible? or is there a way to calculate the coefficient?

Physicist's "$$1/2\otimes 1/2= 1\oplus0$$" is $$2 \otimes 2=3\oplus 1$$ in representation theory language. And "$$1\otimes 0$$" is simply $$3\otimes 1= 3$$.
Clebsch-Gordan coefficient: $$\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|J\,M\rangle$$ is just 1.
If you take the tensor product of a representation with $$j_1 = 1$$ and a representation with $$j_2 = 0$$, the resulting representation is isomorphic to the $$j_1$$ representation. This comes about because the $$j_2 = 0$$ representation, also called the scalar representation, can be just multiplied to the vector. In formulas: pick a vector $$v$$ and a scalar $$s$$, then we can identify $$s \otimes v$$ with $$s v \ .$$