# The Unapologetic Mathematician

## Young Tabloids

Close cousins to Young tableaux, Young tabloids give us another set on which our symmetric group will act.

We say that two Young tableaux are “row-equivalent” if they contain the same entries in the same rows. That is, if we start with a Young tableau and shuffle the entries in each row — but never send an entry from one row to another row — then the resulting tableau is row-equivalent to the one we started with. Any two row-equivalent tableaux are related in this way.

We define a Young tabloid to be a row-equivalence class of Young tableaux, and we write it by writing down any tableau in the class, but with horizontal bars through it. As an example, there are three Young tabloids of shape $(2,1)$:

\displaystyle\begin{aligned}\begin{array}{cc}\cline{1-2}1&2\\\cline{1-2}3&\\\cline{1-1}\end{array}&=\left\{\begin{array}{cc}1&2\\3&\end{array},\begin{array}{cc}2&1\\3&\end{array}\right\}\\\begin{array}{cc}\cline{1-2}1&3\\\cline{1-2}2&\\\cline{1-1}\end{array}&=\left\{\begin{array}{cc}1&3\\2&\end{array},\begin{array}{cc}3&1\\2&\end{array}\right\}\\\begin{array}{cc}\cline{1-2}2&3\\\cline{1-2}1&\\\cline{1-1}\end{array}&=\left\{\begin{array}{cc}2&3\\1&\end{array},\begin{array}{cc}3&2\\1&\end{array}\right\}\end{aligned}

If we have written the tableau abstractly as $t$, then the corresponding tabloid is $\{t\}$ — the equivalence class of $t$.

December 11, 2010 -

1. […] introduced Young tableaux and Young tabloids. We’ve also said that they carry symmetric group actions, but we never really said what they […]

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2. […] that we have an action of on the Young tabloids of shape , we can consider the permutation representation that corresponds to it. Let’s […]

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3. […] corresponding to the partition . For a permutation , the character value is the number of Young tabloids such that . This might be a little difficult to count on its face, but let’s analyze it a […]

Pingback by Characters of Young Tabloid Modules (first pass) « The Unapologetic Mathematician | December 15, 2010 | Reply

4. […] were stymied. But at least we can calculate their dimensions. The dimension of is the number of Young tabloids of shape […]

Pingback by Dimensions of Young Tabloid Modules « The Unapologetic Mathematician | December 16, 2010 | Reply

5. […] leave every entry in on the row where it started. Clearly, this is the stabilizer subgroup of the Young tabloid […]

Pingback by Row- and Column-Stabilizers « The Unapologetic Mathematician | December 22, 2010 | Reply

6. […] if its rows and columns are all increasing sequences. In this case, we also say that the Young tabloid and the polytabloid are […]

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7. […] notions of Ferrers diagrams and Young tableaux, and Young tabloids carry over right away to compositions. For instance, the Ferrers diagram of the composition […]

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8. […] is a Young tabloid with shape , we can define tabloids for each from to by letting be formed by the entries in […]

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9. […] Dominance Lemma for Tabloids If , and appears in a lower row than in the Young tabloid , then dominates . That is, swapping two entries of so as to move the lower number to a higher […]

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10. […] , and use it to build the “column tabloid” . This is defined just like our other tabloids, except by shuffling columns instead of […]

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