# The Unapologetic Mathematician

## 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.