Young’s Natural Representation
Now that we have a canonical basis for our Specht modules composed of standard polytabloids it gives us a matrix representation of for each . We really only need to come up with matrices for the swaps , for , since these generate the whole symmetric group.
When we calculate the action of the swap on a polytabloid associated with a standard Young tableau , there are three possibilities. Either and are in the same column of , they’re in the same row of , or they’re not in the same row or column of .
The first case is easy. If and are in the same column of , then , and thus .
The third case isn’t much harder, although it’s subtler. I say that if and share neither a row nor a column, then is again standard. Indeed, swapping the two can’t introduce either a row or a column descent. The entries to the left of and above are all less than , and none of them are , so they’re all less than as well. Similarly, all the entries to the right of and below are greater than , and none of them are , so they’re all greater than as well.
Where things get complicated is when and share a row. But then they have to be next to each other, and the swap introduces a row descent between them. We can then use our Garnir elements to write this polytabloid in terms of standard ones.
Let’s work this out explicitly for the Specht module , which should give us our well-known two-dimensional representation of . The basis consists of the polytabloids associated to these two tableaux:
We need to come up with matrices for the two swaps and . And the second one is easy: it just swaps these two tableaux! Thus we get the matrix
The action of on the second standard tableau is similarly easy. Since and are in the same column, the swap acts by multiplying by . Thus we can write down a column of the matrix
As for the action on the first tableau, the swap induces a row descent. We use a Garnir element to straighten it out. With the same abuse of notation as last time, we write
and so we can fill in the other column:
From here we can write all the other matrices in the representation as products of these two.
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