## The Dominance Order on Tabloids

Sorry, this should have gone up last Friday.

If is a Young tabloid with shape , we can define tabloids for each from to by letting be formed by the entries in less than or equal to . We define to be the shape of as a composition. For example, if we have

then we define

Along the way we see why we might want to consider a composition like with a zero part.

Anyway, now we define a dominance order on tabloids. If and are two tabloids with composition sequences and , respectively, then we say “dominates” — and we write — if dominates for all .

As a (big!) example, we can write down the dominance order on all tabloids of shape :

It’s an exercise to verify that these are indeed all the tabloids with this shape. For each arrow, we can verify the dominance. As an example, let’s show that

First, let’s write down their composition sequences:

Now it should be easy to see on each row that . As another example, let’s try to compare and . Again, we write down their composition sequences:

We see that , but . Thus neither tabloid dominates the other. The other examples to verify this diagram are all similarly straightforward.

[…] , 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 […]

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[…] of Standard Tableaux Standard tableaux have a certain maximality property with respect to the dominance order on tabloids. Specifically, if is standard and appears as a summand in the polytabloid , then […]

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[…] now we can define the dominance order on column tabloids just like the dominance order for row tabloids. Of course, in doing so we have to alter our definition of the dominance order on […]

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[…] 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) […]

Pingback by Dominance for Generalized Tabloids « The Unapologetic Mathematician | February 9, 2011 |