## Row- and Column-Stabilizers

Every Young tableau with shape gives us two subgroups of , the “row-stabilizer” and the “column-stabilizer” . These are simple enough to define, but to write them succinctly takes a little added flexibility to our notation.

Given a set , we’ll write for the group of permutations of that set. For instance, the permutations that only mix up the elements of the set make up

Now, let’s say we have a tableau with rows . Any permutation that just mixes up elements of leaves all but the first row alone when acting on . Since it leaves every element on the row where it started, we say that it stabilizes the rows of . These permutations form the subgroup . Of course, there’s nothing special about here; the subgroups also stabilize the rows of . And since entries from two different subgroups commute, we’re dealing with the direct product:

We say that is the row-stabilizer subgroup, since it consists of all the permutations that leave every entry in on the row where it started. Clearly, this is the stabilizer subgroup of the Young tabloid .

The column-stabilizer is defined similarly. If has columns , then we define the column-stabilizer subgroup

Now column-stabilizers *do* act nontrivially on the tabloid . The interaction between rearranging rows and columns of tableaux will give us the representations of we’re looking for.

[…] is the column-stabilizer of a Young tableau . If has columns , then . Letting run over is the same as letting run over […]

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[…] every entry is in the same column, the column-stabilizer is all of . And so we calculate the […]

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[…] row of . They cannot be in the same column of , since if they were then the swap would be in the column-stabilizer . Then we could conclude that , which we assumed not to be the case. But if no two entries from the […]

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[…] a tableau , consider the column stabilizer , and use it to build the “column tabloid” . This is defined just like our other […]

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