How to use Sage to find irreducible representation of symmetric group

Question

I am not sure if Sage can be used to find the irreducible representations of the symmetric group.

For example:

For $g = (123)$, we have $$D(a)=\begin{bmatrix}0&0&1\\ 1 & 0 & 0\\0 & 1 & 0\end{bmatrix}$$

One decomposition is the following:
$$D(a)=\begin{bmatrix}1&0&0\\ 0 & -\frac{1}{2} & -\frac{\sqrt{3}}{2}\\0 & \frac{\sqrt{3}}{2} & -\frac{1}{2}\end{bmatrix}$$

so we get two irreducible representations: $$1,\ \ \begin{bmatrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2} & -\frac{1}{2}\end{bmatrix}$$

The answer is not unique since we can pick another similar transformation with different basis.

Can Sage be used to do this for the general case $S_n$? If not, any software can make it?

Answer 1

Symmetric group representations are implemented in Sage.

See the documentation:

• Sage documentation: symmetric group representations