Given $k$ vector spaces $V_1,\dots,V_k$ one can define the tensor product $V_1\otimes\cdots \otimes V_k$ by means of the universal property: it allows any multilinear map $g : V_1\times\cdots\times V_k\to W$ to be written as
$$g(v_1,\dots,v_k)=f(v_1\otimes\cdots\otimes v_k)$$
for a linear $f$. Furthermore, it can be shown that $V^{\otimes k} $ can be identified with the space of multilinear functions $f : V^\ast\times\cdots\times V^\ast\to \mathbb{K}$ while ${V^\ast}^{\otimes k}$ can be identified withConsider the spacefollowing two understandings of all multilinear functions $f : V\times\cdots\times V\to \mathbb{K}$.
This allows tensors to be presented usually just as follows: a tensor of type $(r,s)$ in the vector space $V$ over $\mathbb{K}$ is a multilinear map $T : V\times\cdots\times V\times V^\ast\times\cdots\times V^\ast \to \mathbb{K}$ where there are $r$ copies of $V$ and $s$ copies of $V^\ast$.
In Quantum Mechanics, on the other hand, tensors seems to be an entirely different thing.
They are somehow related to the representations of the rotation group $SO(3)$. Furthermore, sometimes one talks about "irreducible tensors". More than that, I've seem for example in Sakurai, the author decomposing
$$U_iV_j=\dfrac{\mathbf{U}\cdot\mathbf{V}}{3}\delta_{ij}+\dfrac{(U_iV_j-U_jV_i)}{2}+\left(\dfrac{U_iV_j+U_jV_i}{2}-\dfrac{\mathbf{U}\cdot\mathbf{V}}{3}\delta_{ij}\right)$$
where the author says that $U_iV_j$ are the components of a reducible tensor which is reduced according to the above formula.
Given $k$ vector spaces $V_1,\dots,V_k$ one can define the tensor product $V_1\otimes\cdots \otimes V_k$ by means of the universal property: it allows any multilinear map $g : V_1\times\cdots\times V_k\to W$ to be written as
$$g(v_1,\dots,v_k)=f(v_1\otimes\cdots\otimes v_k)$$
for a linear $f$. Furthermore, it can be shown that $V^{\otimes k} $ can be identified with the space of multilinear functions $f : V^\ast\times\cdots\times V^\ast\to \mathbb{K}$ while ${V^\ast}^{\otimes k}$ can be identified with the space of all multilinear functions $f : V\times\cdots\times V\to \mathbb{K}$.
This allows tensors to be presented usually just as follows: a tensor of type $(r,s)$ in the vector space $V$ over $\mathbb{K}$ is a multilinear map $T : V\times\cdots\times V\times V^\ast\times\cdots\times V^\ast \to \mathbb{K}$ where there are $r$ copies of $V$ and $s$ copies of $V^\ast$.
In Quantum Mechanics, on the other hand, tensors seems to be an entirely different thing.
They are somehow related to the representations of the rotation group $SO(3)$. Furthermore, sometimes one talks about "irreducible tensors". More than that, I've seem for example in Sakurai, the author decomposing
$$U_iV_j=\dfrac{\mathbf{U}\cdot\mathbf{V}}{3}\delta_{ij}+\dfrac{(U_iV_j-U_jV_i)}{2}+\left(\dfrac{U_iV_j+U_jV_i}{2}-\dfrac{\mathbf{U}\cdot\mathbf{V}}{3}\delta_{ij}\right)$$
where the author says that $U_iV_j$ are the components of a reducible tensor which is reduced according to the above formula.
This doesn't seem to match the usual definitions of representation theory, where a representation of a group $G$ is a pair $(\rho,V)$ with $\rho : G\to GL(V)$ a homomorphism and where such representation is irreducible when there is no proper invariant subspace. In other words, representations can be reducible or irreducible, not tensors.
To make things worse there are the so-called spherical tensors which are objects $T_k^q$ with two indices and that are somehow related to the irreducible representations of $SO(3)$. It seems all tensors in QM are of this type.
But again, in the standard lore, tensors can have lots of indices (see for example the Riemann curvature tensor in Differential Geometry, with four indices).
So there seems to be a bunch of things mixed together and I'm unable to understand what is actually going on here.
My doubts are:
How are tensors from QM and tensors from linear algebra widely used in geometry are related?
What do tensors have to do with representations of the rotation group anyway?
What is the meaning of reducibiityreducibility of a tensor opposed to the reducibility of a representation of a group?
Why these spherical tensors seem to be all tensors in QM and how do they relate to the usual tensors too?
