Transformations under some representation

Question

I have a profound question. We call spinor an element of a space which transforms with the irreps with semi-integer $j$ of $SU(2)$. But i can't find the meaning of this. How could it transform in other ways? I mean, i am working with $SU(2)$ matrices so it MUST transfom with them. If i use a random matrix it would transform in other ways, so what's the meaning? A tensor is defined by how it transforms under some action of a group, but how could it transform in other ways if we apply only that group?

In 3D space we impose the distance to be not-modified, so we found the group of transformation is $SO(3)$. Why we say "vectors are quantities that transforms with $SO(3)$ matrices"? I can use a general matrix and it would be still a coordinate in the space.

For example Lorentz group, we impose the $s^2$ to rest the same, so we get transformations left-right spinors like $(1/2,0)$ or $(1,1)$ but what does this all mean?

Answer 1

1. In the context of $SU(2)$, it is only the $j=\frac{1}{2}$ representation that is called a spinor representation. It is also called the fundamental or defining representation. Let's call it $V$. It is 2-dimensional.

   The higher irreducible representations $j\in\frac{1}{2}\mathbb{N}$ with dimension $2j+1$ can in turn be realized as the $2j$'th symmetric tensor product $V^{\odot 2j}$, cf. e.g. Ref. 1.

2. That $SU(2)$ is the double cover of $SO(3)$ is discussed in e.g. this, this, this & this Phys.SE posts.

3. For irreducible representations of the Lorentz group, see e.g. this Phys.SE post.

References:

1. G. 't Hooft, Introduction to Lie Groups in Physics, lecture notes, p. 35-37. The pdf file is available here.