The Dominance Lemma for Tabloids

If $k<l$ , and $k$ appears in a lower row than $l$ in the Young tabloid $\{t\}$ , then $(k\,l)\{t\}$ dominates $\{t\}$ . That is, swapping two entries of $\{t\}$ so as to move the lower number to a higher row moves the tabloid up in the dominance relations.

Let the composition sequences of $\{t\}$ and $(k\,l)\{t\}$ be $\lambda^i$ and $\mu^i$ , respectively. For $i<k$ and $i\geq l$ we automatically have $\lambda^i=\mu^i$ . For $k\leq i<l$ there is a difference between the two: the entry $k$ has been added in a different place. Let $k$ and $l$ be in rows $q$ and $r$ of $\{t\}$ , respectively. In $\lambda^i$ , the entry $k$ is added to row $q$ , while in $\mu^i$ it’s been added to row $r$ . That is, $\lambda^i$ is the same as $\mu^i$ with part $q$ increased by one and part $r$ decreased by one. Our assumption that $k$ is in a lower row than $l$ in $\{t\}$ is that $q>r$ . Therefore, since the lower row in $\lambda^i$ is less than in $\mu^i$ , we find that $\lambda^i\triangleleft\mu^i$ . And we conclude that $\{t\}\trianglelefteq(k\,l)\{t\}$ , as asserted.

January 11, 2011 - Posted by John Armstrong | Algebra, Representation Theory, Representations of Symmetric Groups

3 Comments »

[…] any such pair, the dominance lemma tells us that . That is, by “untwisting” the column inversion, we can move up the […]

Pingback by The Maximality of Standard Tableaux « The Unapologetic Mathematician | January 13, 2011 | Reply
[…] 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 , and appears in a column to the right of in the column […]

Pingback by The Column Dominance Order « The Unapologetic Mathematician | January 20, 2011 | Reply
[…] of course have a dominance lemma: if , occurs in a column to the left of in , and is obtained from by swapping these two […]

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

« Dec Feb »

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
Search for:

The Unapologetic Mathematician

Mathematics for the interested outsider