I'm trying to get some group theory basics from Wu-Ki Tung, "Group Theory in Physics". I'm having some trouble with his definitions of direct and tensor products of representations and representation spaces.
He defines
"the direct product space $U \times V$ consists of all linear combinations of the orthonormal basis vectors $\{\hat{w}_k; k = (i, j); i = 1,...,n_\mu; j = 1,...,n_\nu\}$ where $\hat{w}_k$, can be regarded as the “formal product” $\hat{w}_k=\hat{u}_i\cdot\hat{v}_j$"
but to me, this looks a lot like a tensor product, since the dimension of $W$ would be $n_\mu n_\nu$ instead of $n_\mu+n_\nu$ that is what it'd like for my direct product (or direct sum, since we're talking about vector spaces).
He then says
"The direct product space $V_m \times V_m \times ... \times V_m$ involving $n$ factors of $V$ shall be referred to as the tensor space and denoted by $V^n_m$"
but don't you need a tensor product to define the tensor space?
Lastly, he says that the direct product representation can be decomposed like $D^{\mu\times\nu}=\bigoplus_\lambda a_\lambda D^\lambda$. This seems correct, but since my two other questions, I'm doubting that I understand this too.
