The sign representation of the Symmetric Group

I am currently trying to learn some of the basics of Representation Theory through Sagan's The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions.

On page 11, after briefly describing the concept of a submodule, he provides an example with $G:=\{g_1,g_2,\ldots,g_n\}$ and $V:=\mathbb{C}[G]$. Furthermore, he proves that $W:=\mathbb{C}[g_1+\dotsb+g_n]$ is a submodule of $V$ on which $G$ acts trivially.

The reader should verify that if $G= S_n$, then the sign representation can also be recovered by using the submodule $$W:=\mathbb{C}\left[\sum_{\pi\in S_n}sgn(\pi)\pi\right].$$

I'm not completely sure what he means by that. I first proved that $W$ is a submodule of $V:=\mathbb{C}[S_n]$ as $$\sum_{\pi\in S_n}sgn(\pi)\sigma\pi =\pm\sum_{\pi\in S_n}sgn(\pi)\pi$$ holds for every $\sigma\in G$.

What would be my next step in order to show that the sign representation can be recovered from this module?

• The claim is that that submodule is isomorphic to the sign representation. Find an isomorphism! Commented Apr 29, 2014 at 7:20
• Thank you for your reply, Mariano. I believe I see what you are saying. Could you tell me if my intuition is correct and if so, help me formalize it? I believe that the isomorphism is $$sgn(\sigma)\mapsto sgn(\sigma)\sum_{\pi\in S_n}sgn(\pi)\pi$$ is this correct? Commented Apr 29, 2014 at 7:31

It suffices to prove that $$\sigma\cdot w=sgn(\sigma)w$$ for all $w\in W$. Indeed, in this case, any vector space isomorphism $W\rightarrow\mathbb{C}$ can be seen to be an isomorphism of $W$ with the sign representation.
Note that $$\sigma\left(\sum_{\pi\in S_n}sgn(\pi)\pi\right)=\sum_{\pi\in S_n}sgn(\pi)\sigma\pi$$ $$=\sum_{\pi\in S_n}sgn(\sigma)sgn(\sigma\pi)\sigma\pi$$ $$=sgn(\sigma)\sum_{\pi\in S_n}sgn(\pi)\pi,$$ as desired.