The Unapologetic Mathematician

Mathematics for the interested outsider

“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:

\displaystyle t=\begin{array}{ccc}1&2&3\\5&4&\\6&&\end{array}

Now, it’s slightly abusive to the notation, but we’ll just write a tableau t and know that we actually mean the polytabloid e_t 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 e_t 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

\displaystyle u=\begin{array}{ccc}1&4&3\\2&5&\\6&&\end{array}

We can write down another table, just like before:

\displaystyle\begin{array}{cccc}A'&B'&\pi&\pi u\\\hline\\\{4,5\}&\{3\}&e&\begin{array}{ccc}1&4&3\\2&5&\\6&&\end{array}\\\{3,5\}&\{4\}&(3\,4)&\begin{array}{ccc}1&3&4\\2&5&\\6&&\end{array}\\\{3,4\}&\{5\}&(3\,5\,4)&\begin{array}{ccc}1&3&5\\2&4&\\6&&\end{array}\end{array}

which lets us write


Similarly we can write




Putting these all together, we conclude that


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

January 25, 2011 - Posted by | Algebra, Representation Theory, Representations of Symmetric Groups

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