Examples of Specht Modules
Let’s look at a few examples of Specht modules.
First, let . The only polytabloid is
on which acts trivially. And so is a one-dimensional space with the trivial group action. This is the only possibility anyway, since , and we’ve seen that is itself a one-dimensional vector space with the trivial action of .
Next, consider — with parts each of size . This time we again have one polytabloid. We fix the Young tableau
Since every entry is in the same column, the column-stabilizer is all of . And so we calculate the polytabloid
We use our relations to calculate
We conclude that is a one-dimensional space with the signum representation of . Unlike our previous example, there is a huge difference between and ; we’ve seen that is actually the left regular representation, which has dimension .
Finally, if , then we can take a tableau and write a tabloid
where the notation we’re using on the right is well-defined since each tabloid is uniquely identified by the single entry in the second row. Now, the polytabloid in this case is , since the only column rearrangement is to swap and . It’s straightforward to see that these polytabloids span the subspace of where the coefficients add up to zero:
As a basis, we can pick . We recognize this pattern from when we calculated the invariant subspaces of the defining representation of . And indeed, is the defining representation of , which contains as the analogous submodule to what we called before.