The Unapologetic Mathematician

Mathematics for the interested outsider

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 | Algebra, Representation Theory, Representations of Symmetric Groups | 3 Comments