The Dominance Lemma for Tabloids
If 
, and 
appears in a lower row than 
in the Young tabloid 
, then 
dominates . That is, swapping two entries of so as to move the lower number to a higher row moves the tabloid up in the dominance relations.
Let the composition sequences of and be 
and 
, respectively. For 
and 
we automatically have 
. For 
there is a difference between the two: the entry has been added in a different place. Let and be in rows 
and 
of , respectively. In , the entry is added to row , while in it’s been added to row . That is, is the same as with part increased by one and part decreased by one. Our assumption that is in a lower row than in is that 
. Therefore, since the lower row in is less than in , we find that 
. And we conclude that 
, as asserted.
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January 11, 2011 -
Posted by John Armstrong |
Algebra, Representation Theory, Representations of Symmetric Groups
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