# 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.

January 20, 2011

## The Column Dominance Order

Okay, for the last couple posts I’ve talked about using Garnir elements to rewrite nonstandard polytabloids — those coming from tableaux containing “row descents” — in terms of “more standard” polytabloids. Finally, we’re going to define another partial order that will give some meaning to this language.

Given a tableau $t$, consider the column stabilizer $C_t$, and use it to build the “column tabloid” $[t]=C_tt$. This is defined just like our other tabloids, except by shuffling columns instead of rows.

For example, consider the tabloid

$\displaystyle t=\begin{array}{cc}1&2\\3&\end{array}$

from which we get the column tabloid

$\displaystyle\begin{array}{|c|c|}1&2\\3&\multicolumn{1}{c}{}\end{array}=\left\{\begin{array}{cc}1&2\\3&\end{array},\begin{array}{cc}3&2\\1&\end{array}\right\}$

And now we can define the dominance order on column tabloids just like the dominance order for row tabloids. Of course, in doing so we have to alter our definition of the dominance order on Ferrers diagrams to take columns into account instead of rows.

But one thing at least will make our life simpler: it should be clear that we still have a dominance lemma for column dominance. To be explicit: if $k< l$, and $k$ appears in a column to the right of $l$ in the column tabloid $[t]$, then $(k\,l)[t]$ dominates $[t]$ in the column dominance order.

January 20, 2011