I'm a physicist trying to get some group theory basics from WKT. 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.