# How does the "Alternating Operator" distribute in Tensors?

(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 $$\mathscr{A}^2 = \mathscr{A}$$.)