The Unapologetic Mathematician

Mathematics for the interested outsider

Row- and Column-Stabilizers

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

Given a set X, we’ll write S_X for the group of permutations of that set. For instance, the permutations that only mix up the elements of the set \{1,2,4\} make up S_{\{1,2,4\}}

Now, let’s say we have a tableau t with rows R_1,\dots,R_k. Any permutation that just mixes up elements of R_1 leaves all but the first row alone when acting on t. Since it leaves every element on the row where it started, we say that it stabilizes the rows of t. These permutations form the subgroup S_{R_1}. Of course, there’s nothing special about R_1 here; the subgroups S_{R_i} also stabilize the rows of t. And since entries from two different subgroups commute, we’re dealing with the direct product:

\displaystyle R_t=S_{R_1}\times\dots\times S_{R_k}

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

The column-stabilizer is defined similarly. If t has columns C_1,\dots,C_l, then we define the column-stabilizer subgroup

\displaystyle C_t=S_{C_1}\times\dots\times S_{C_l}

Now column-stabilizers do act nontrivially on the tabloid \{t\}. The interaction between rearranging rows and columns of tableaux will give us the representations of S_n we’re looking for.

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


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