representations of s3

Last updated January 27, 2022

Idea

The general idea is the following:

1. Consider the abelian subgroup $A_3\subset S_3$ generated by some 3-cycle $\tau$.
2. The irreducible representations of a finite abelian group are 1-dimensional, hence we can decompose the representation $W$ of $A_3$ into a direct sum of spaces spanned by each of the three eigenvectors. $$W=\bigoplus V_i$$
3. To consider the rest of $S_3$, it suffices to consider a transposition $\sigma$ (since $\tau$ and $\sigma$ generate $S_3$). It turns out that $\sigma$ just permutes the components of the decomposition above, e.g. eigenvectors are sent to eigenvectors.