Young tableaux to Specht polynomial to Irreducible representation for $(1,3,5) \in S_5$

Question

What I am trying to do?

Work out the irreducible representation of the group element $(1,3,5) \in S_5$ for the partition $2+2+1$ .

Motivation:

Learn how to calculate irreducible representation from Young tableaux.

Group: $S_5$

Partitions and Young tableaux:

Now I pick the partition $2+2+1$ and work out the representation of the group element $(1,3,5) \in S_5$.

First I work out the standard Young tableaux for the shape $2+2+1$.

Now I compute the Specht polynomials for each of the tableau.

Now I compute the representation for the permutation $\left(1, 3, 5\right) \in S_5$.

So, the irreducible representation of $\left(1, 3, 5\right)$ is :

My question: Am I doing it right?