I am trying to draw the Young diagram for $S_5$. I know the following pieces of information about $S_5$.
- The order of the group is $120$.
- The number of conjugacy classes and so partitions is $7$.
- Degrees of irreducible representations $1,1,4,4,5,5,6$.
- The partition is $1 + 10 + 15 + 20 + 20 + 24 + 30 = 120$.
I understand that the Young diagram should contain $30$ boxes in the first row, $24$ boxes in the second row, $20$ boxes in the third and fourth rows, $15$ boxes in the fifth row, $10$ boxes in the sixth row and $1$ box in the seventh row.
So, the Young diagram is as follows.
Am I doing it right? I understand that the next step is to fill up the boxes to make it a Young tableau.
I was able to compute the partition as follows.
What should be my next step?
I think I was able to draw the Young diagrams.