The Unapologetic Mathematician

Mathematics for the interested outsider

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 Posted by | Algebra, Group theory, Representation Theory, Representations of Symmetric Groups | 13 Comments

   

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