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.
[…] any such pair, the dominance lemma tells us that . That is, by “untwisting” the column inversion, we can move up the […]
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[…] 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 […]
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[…] of course have a dominance lemma: if , occurs in a column to the left of in , and is obtained from by swapping these two […]
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