Outer Tensor Products Let’s say we have two finite groups and ,
and we have (left) representations of each one: and . It turns
out that the tensor product naturally carries a representation
of the product group . Equivalently, it carries a representation
of each of and , and these representations commute with each
other. In our module notation, we write . The action is simple
enough. Any vector in can be written (not always uniquely) as a
sum of vectors of the form . We let act on the first
“tensorand”, let act on the second, and extend by linearity.
That is: and the action of either or on the sum of two tensors
is the sum of their actions on each of the tensors. Now, might
the way we write the sum make a difference? No, because all the
relations look like where in the last equation is a complex
constant. Now, we can check that the actions of and give
equivalent results on either side of each equation. For
instance, acting by in the first equation we see just as we
want. All the other relations are easy enoug