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 - Posted by John Armstrong | Algebra, Representation Theory, Representations of Symmetric Groups

2 Comments »

[…] Properties of Garnir Elements from Tableaux 2 When we pick a tableau with a certain row descent and use it to pick sets and , 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 corresponding to the original tableau is dominated by all the other tabloids, using the column dominance order. […]

Pingback by Properties of Garnir Elements from Tableaux 2 « The Unapologetic Mathematician | January 20, 2011 | Reply
[…] this: we’re going to have orders on generalized tabloids analogous to the dominance and column dominance orders for tabloids without repetitions. Each tabloid (or column tabloid) gives a sequence of […]

Pingback by Dominance for Generalized Tabloids « The Unapologetic Mathematician | February 9, 2011 | Reply

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.

RSS Feeds

RSS - Posts
RSS - Comments
Feedback

Got something to say? Anonymous questions, comments, and suggestions at Formspring.me!
Subjects

Subjects
Archives

January 2011

M T W T F S S

1 2

3 4 5 6 7 8 9

10 11 12 13 14 15 16

17 18 19 20 21 22 23

24 25 26 27 28 29 30

31

« Dec Feb »
Search for:

The Unapologetic Mathematician

Mathematics for the interested outsider