# 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 $t^\lambda$.