# The Unapologetic Mathematician

## Properties of Garnir Elements from Tableaux 2

When we pick a tableau $t$ with a certain row descent and use it to pick sets $A$ and $B$, as we’ve done, the resulting Garnir element is a sum of a bunch of tabloids coming from a bunch of tableaux. I say that the column tabloid $[t]$ corresponding to the original tableau is dominated by all the other tabloids, using the column dominance order.

Indeed, when considering column tabloids we can rearrange the entries within columns freely, so we may assume that they’re always increasing down the columns. If we have our row descent in row $i$, we can label the entries in the left column by $a$s and those in the right column by $b$s. Our tabloid then looks — in these two columns, at least — something like

$\displaystyle\begin{array}{ccc}a_1&\hphantom{X}&b_1\\&&\wedge\\a_2&&b_2\\&&\wedge\\\vdots&&\vdots\\&&\wedge\\a_i&>&b_i\\\wedge&&\\\vdots&&\vdots\\\wedge&&b_q\\a_p&&\end{array}$

We see our sets $A=\{a_i,\dots,a_p\}$ and $B=\{b_1,\dots,b_i\}$. The permutations in the transversal that we use to construct our Garnir element work by moving swapping some of the $b$s with some of the $a$s. But since all that $b$s are smaller than all the $a$s, while they occur in a row further to the right, the dominance lemma for column tabloids tells us that any such swap can only move the tabloid up in the dominance order.

It is in this sense that the Garnir element lets us replace a tabloid with a linear combination of other tabloids that are “more standard”. And it puts us within striking distance of our goal.