# Direct product and tensor product in representation theory

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.

• 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 Jan 8, 2021 at 16:33
• 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. Jan 8, 2021 at 16:51