1. ## Linear representations

Prove there exists a left ideal I of C[$\displaystyle S_{3}$] (so the ring in the symmetric group S3 with complex coeffs) which does not contain the element $\displaystyle e_{1}$:= $\displaystyle \frac{1}{6}$ $\displaystyle \sum$ [y] nor the element $\displaystyle e_{2}$:= $\displaystyle \frac{1}{6}$ $\displaystyle \sum$ sgn(y) [y]

(Where both the sums are over all $\displaystyle y$ in $\displaystyle S_{3}$)

(Forgive my Latex; I'm still trying to get the hang of it)

2. Originally Posted by brisbane
Prove there exists a left ideal I of C[$\displaystyle S_{3}$] (so the ring in the symmetric group S3 with complex coeffs) which does not contain the element $\displaystyle e_{1}$:= $\displaystyle \frac{1}{6}$ $\displaystyle \sum$ [y] nor the element $\displaystyle e_{2}$:= $\displaystyle \frac{1}{6}$ $\displaystyle \sum$ sgn(y) [y]

(Where both the sums are over all $\displaystyle y$ in $\displaystyle S_{3}$)

(Forgive my Latex; I'm still trying to get the hang of it)
Consider $\displaystyle M=\mathbb{C}[S_3]$ as a $\displaystyle \mathbb{C}[S_3]$-module of itself.

Then, M is decomposed as (Dummit & Foote p869-870)

$\displaystyle M = z_1M \oplus z_2M \oplus \cdots \oplus z_rM$, where $\displaystyle z_i = \frac{\chi_i(1)}{|S_3|}\sum_{g \in S_3}\chi_i(g^{-1})g$.

Each $\displaystyle z_iM$ is a submodule of M (two-sided ideal of M) and your $\displaystyle e_1, e_2$ correspond to $\displaystyle z_1, z_2$, respectively.

Since $\displaystyle S_3$ has three conjugacy classes, there are three irreducible characters of $\displaystyle \mathbb{C}[S_3]$.
The $\displaystyle z_3M$ is the ideal you are looking for, where $\displaystyle z_3=\frac{2}{6}(2e - (123) - (132))$.

Note that $\displaystyle z_3M$ has a dimension 4, which is not irreducible.