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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.

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    $\begingroup$ Related/possible duplicate: physics.stackexchange.com/q/447342/50583 Some physicists are really bad at keeping the mathematical notions of direct product and tensor product straight (presumably because the natural representation of the direct product of groups $G\times H$ is the tensor product of representation spaces $V_G\otimes V_H$. See also Qmechanic's answer at physics.stackexchange.com/a/206849/50583 $\endgroup$
    – ACuriousMind
    Commented Jan 8, 2021 at 16:33
  • $\begingroup$ Your question is a duplicate of several questions linked above. Most physicists use direct product for tensor product, instead of Cartesian product, as they deal with vector spaces and linear operators/representations of both Lie algebras and exponentials of their elements. Cf Kronecker product. $\endgroup$ Commented Jan 8, 2021 at 16:51

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