# Thread: Linear representations

1. ## Linear representations

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

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

(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[ $S_{3}$] (so the ring in the symmetric group S3 with complex coeffs) which does not contain the element $e_{1}$:= $\frac{1}{6}$ $\sum$ [y] nor the element $e_{2}$:= $\frac{1}{6}$ $\sum$ sgn(y) [y]

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

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

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

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

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

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

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