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.
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December 22, 2010 -
Posted by John Armstrong |
Algebra, Representation Theory, Representations of Symmetric Groups
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