# The Unapologetic Mathematician

## 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 -

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

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3. [...] 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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4. [...] 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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