# Show that the natural representation of S3 is a direct sum of irreps

1. Oct 17, 2013

1. The problem statement, all variables and given/known data
Hey everyone!

So to elaborate the title a bit more: basically I have to show that the natural representation of $S_{3}$ is a direct sum of the one-dimensional irreducible representation and the two-dimensional irreducible representation of $S_{3}$.

2. Relevant equations
Im not sure if there are any...

3. The attempt at a solution
So I dont have any formal attempt at the solution because I dont know where to start, so some explicit help on that would be great. if you could please address the following things in particular:

- what is a natural representation - it it just a representation in which the basis vectors are simply columns of unit length in the direction they point? like $e_{1},e_{2},e_{3}$ etc?

- Is the 1-D irrep just the trivial rep?

- what is the 2-D irrep?

Some help would be great guys, thanks!