I'm reading through my notes on representation theory of $S_n$ and $GL(V)$, and have come unstuck on a definition which I can't understand - furthermore I can't seem to find any information on it online, so I can't read up on it to figure out what's going on. I'd be very grateful for an explanation or some direction to somewhere online I can find out more about it. Our notes contain the following definition:
Suppose $V$ is an $m$-dimensional vector space over $\mathbb{C}$, and $n \in \mathbb{Z}$: then we denote the 1-dimensional $\mathbb{C}GL(V)$ module corresponding to the representation $GL(V) \to \mathbb{C}^*$, $g \to (\det g)^n$ is denoted $\det ^n$. So, for example, $\det ^ 1 \cong \Lambda^mV$, where $\Lambda^m V$ denotes the $m$-th exterior power of V, $\det^n \cong (\det^1 )^{\otimes n}$ if $n \geq 0$ and $\det^n \cong (\det^{-n})^*$ if $n \leq 0$.
So, I don't really get what this module actually is: I know what the exterior powers are, but why is it that the module described is equal to the exterior power for $n = 1$, and equal to the other objects stated for other values of $n$? I can't find anything useful on these "det modules", so if anyone could explain step-by-step what's being said here, that would be extremely helpful. Obviously I have the basic representation theory background (what tensor products, $\mathbb{C}GL(V)$ are etc), but if you could keep explanations relatively simple here if possible I'd be very grateful. Many thanks in advance.
(Edit: just in case it isn't possible to tell what these modules are from my given portion of the notes, they later go on to prove "the 1-dimensional rational $\mathbb{C}GL(V)$-modules are precisely the $\det^n,\,n \in \mathbb{Z}$" if that helps provide some context.)