## “Straightening” a Polytabloid

Let’s look at one example of “straightening” out a polytabloid to show it’s in the span of the standard polytabloids, using the Garnir elements.

We’ll start with one we’ve already partially worked out:

Now, it’s slightly abusive to the notation, but we’ll just write a tableau and know that we actually mean the polytabloid in our linear combinations. Using this, we’ve seen that we can write

Now, by the way we’ve selected our Garnir elements, we know that none of these can have any column descents. And we also know that they can’t have a row descent in the same place did. Indeed, the only three that have a row descent all have it between the second and third entries of the first row. So now let’s look at

We can write down another table, just like before:

which lets us write

Similarly we can write

and

Putting these all together, we conclude that

All of these tabloids are standard, and so we see that our original — nonstandard — is in the span of the standard polytabloids.