(I'm not sure that I even phrased the question correctly. I will explain more about this below.)
Given a k-tensor $T$, we can define an alternating k-tensor $Alt(T)$ in the following way:
where $\epsilon$ is the sign function.
My first question is, what do we even call the $Alt$? In my title, I called it the "Alternating Operator". I would like to know its formal name.
And here's my main question. Given alternating k- and l- tensors $\theta, \eta$, we have:
It seems that the "Alternating Operator" distributes to the two tensors in the first equality. Why is this?
I am new to tensors (just starting studying this month). I am aware that the tensor product distributes, but is this in any way related to the reason why the Alternating Operator also distributes? If so, how?
I think if you call it the "Alternating operator" people will generally understand what you mean. But at least in my circles it is more frequently called "the (total) antisymmetrization of $T$".
For your main question: the operator $\mathscr{A}$ is a linear mapping from $\mathscr{T}^k$ to itself. For any linear mapping you have the distribution law $L(x - y) = L(x) - L(y)$.
(It is in fact not just a linear mapping but a projection, meaning that $\mathscr{A}^2 = \mathscr{A}$.)