## 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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[...] would make the zero map. So among the nonzero , there are some with maximal in the column dominance order. I say that we can find a semistandard among [...]

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