Dominance for Generalized Tabloids
Sorry I forgot to post this yesterday afternoon.
You could probably have predicted 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 compositions, and at the th step we throw in all the entries with value
.
For example, the generalized column tabloid
gives the sequence of compositions
while the semistandard generalized column tabloid
gives the sequence of compositions
and we find that since
for all
.
We of course have a dominance lemma: if ,
occurs in a column to the left of
in
, and
is obtained from
by swapping these two entries, then
. As an immediate corollary, we find that if
is semistandard and
is different from
, then
. That is,
is the "largest" (in the dominance order) equivalence class in
. The proofs of these facts are almost exactly as they were before.
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