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.