Show that a submodule is isomorphic to the two-dimensional irreducible subrepresentation of the standard permutation representation of $S_3$ let $U$ be the submodule of of $\mathbb{C}[\mathfrak{S}_3]$ generated by $\bar{u} := 1-(12)+(23)-(123)$. 
Show that $U$ is isomorphic to the two-dimensional irreducible subrepresentation of thestandard permutation representation of $\mathfrak{S}_3$. 
How does one go about showing that a submodule is isomorphic to the representation?
 A: Consider $W=\mathrm{span}(\overline{u},\overline{v})$, where $\overline{v}=(23)\overline{u}$. Then, $(12)\overline{u}=-\overline{u}$ and $(12)\overline{v}=-\overline{u}+\overline{v}$, and $(23)\overline{u}=\overline{v}$ and $(23)\overline{v}=\overline{u}$.
Since $W$ is closed under the action of the generators of $S_3$, $W=\mathbb{C}S_3.\overline{u}$, and is, therefore, a $2$-dimensional module. 
To show that $W$ is irreducible, note that any nontrivial proper submodule must be $1$-dimensional.
Therefore, assume $x=a\overline{u}+b\overline{v}$ spans a $1$-dimensional submodule. Since such a module is closed under the action of $(12)$ and $(23)$, we must have $(12)x=\lambda x$ and $(23)x=\mu x$ for some $\lambda,\mu\in\mathbb{C}$. Now, compute
$$
\lambda a\overline{u}+\lambda b\overline{v}=\lambda x=(12)x=-(a+b)\overline{u}+b\overline{v}.
$$
Comparing the coefficients of $\overline{v}$, we have that $\lambda b=b$, so $\lambda=1$. Comparing the coefficients of $\overline{u}$, we see $a=-(a+b)$, so $b=-2a$.
Next, compute
$$
\mu a\overline{u}+\mu b\overline{v}=\mu x=(23)x= b\overline{u}+a\overline{v}
$$
so $\mu a=b$ and $\mu b=a$. It follows that $\mu^2=1$, so $a= b$ or $a=-b$. If $a=b$, then, by the computation above, $a=-2a$, so $a=b=0$. But, we get that same conclusion if $b=-a$, since then $a=2a$. Hence, $x=0$ spans a $0$-dimensional submodule.
Since no $1$-dimensional submodule exists, it follows that $W$ is irreducible. It must therefore be isomorphic to the unique irreducible $S_3$-module.
More explicitly, if we take the basis $\{e_1-e_2,e_2-e_3\}$ for the reflection representation (inside $\mathbb{C}^3$), then $\overline{u}\mapsto e_1-e_2$ and $\overline{v}\mapsto e_1-e_3=(e_1-e_2)+(e_2-e_3)$ defines the isomorphism.
