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.