# The Unapologetic Mathematician

## Young Tableaux

We want to come up with some nice sets for our symmetric group to act on. Our first step in this direction is to define a “Young tableau”.

If $\lambda\vdash n$ is a partition of $n$, we define a Young tableau of shape $\lambda$ to be an array of numbers. We start with the Ferrers diagram of the partition $\lambda$, and we replace the dots with the numbers $1$ to $n$ in any order. Clearly, there are $n!$ Young tableaux of shape $\lambda$ if $\lambda\vdash n$.

For example, if $\lambda=(2,1)$, the Ferrers diagram is $\displaystyle\begin{array}{cc}\bullet&\bullet\\\bullet&\end{array}$

We see that $(2,1)\vdash3$, and so there are $3!=6$ Young tableaux of shape $(2,1)$. They are \displaystyle\begin{aligned}\begin{array}{cc}1&2\\3&\end{array}&,&\begin{array}{cc}1&3\\2&\end{array}&,&\begin{array}{cc}2&1\\3&\end{array}\\\begin{array}{cc}2&3\\1&\end{array}&,&\begin{array}{cc}3&1\\2&\end{array}&,&\begin{array}{cc}3&2\\1&\end{array}\end{aligned}

We write $t_{i,j}$ for the entry in the $(i,j)$ place. For example, the last tableau above has $t_{1,1}=3$, $t_{1,2}=2$, and $t_{2,1}=1$.

We also call a Young tableau $t$ of shape $\lambda$ a “ $\lambda$-tableau”, and we write $\mathrm{sh}(t)=\lambda$. We can write a generic $\lambda$-tableau as $t^\lambda$.

December 9, 2010 -

1. […] cousins to Young tableaux, Young tabloids give us another set on which our symmetric group will […]

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2. […] Action on Tableaux and Tabloids We’ve introduced Young tableaux and Young tabloids. We’ve also said that they carry symmetric group actions, but we never […]

Pingback by The Action on Tableaux and Tabloids « The Unapologetic Mathematician | December 13, 2010 | Reply

3. […] Young tableau thus contains all numbers on the single row, so they’re all row-equivalent. There is only […]

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4. […] and be Young tableaux of shape and , respectively. If for each row, all the entries on that row of are in different […]

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5. […] Young tableau with shape gives us two subgroups of , the “row-stabilizer” and the […]

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

6. […] is the column-stabilizer of a Young tableau . If has columns , then . Letting run over is the same as letting run over for each from to […]

Pingback by Polytabloids « The Unapologetic Mathematician | December 23, 2010 | Reply

7. […] is the submodule of the Young tabloid module spanned by the polytabloids where runs over the Young tableaux of shape […]

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8. […] let and are two Young tableaux of shapes and , respectively, where and . If — where is the group algebra element […]

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9. […] say that a Young tableau is “standard” if its rows and columns are all increasing sequences. In this case, we […]

Pingback by Standard Tableaux « The Unapologetic Mathematician | January 5, 2011 | Reply

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

Pingback by Compositions « The Unapologetic Mathematician | January 6, 2011 | Reply

11. […] predictably enough, certain Garnir elements we’re particularly interested in. These come from Young tableaux, and will be useful to us as we move […]

Pingback by Garnir Elements from Tableaux « The Unapologetic Mathematician | January 17, 2011 | Reply

12. […] as a quick use of this concept, think about how to fill a Ferrers diagram to make a standard Young tableau. It should be clear that since is the largest entry in the tableau, it must be in the rightmost […]

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13. […] Young Tableaux And now we have another generalization of Young tableaux. These are the same, except now we allow repetitions of the […]

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