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.

About these ads

December 9, 2010 - Posted by | Algebra, Group theory, Representation Theory, Representations of Symmetric Groups

13 Comments »

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

    Pingback by Young Tabloids « The Unapologetic Mathematician | December 11, 2010 | Reply

  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 [...]

    Pingback by Permutation Representations from Partitions « The Unapologetic Mathematician | December 14, 2010 | Reply

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

    Pingback by The Dominance Lemma « The Unapologetic Mathematician | December 20, 2010 | Reply

  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 [...]

    Pingback by Specht Modules « The Unapologetic Mathematician | December 27, 2010 | Reply

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

    Pingback by Corollaries of the Sign Lemma « The Unapologetic Mathematician | December 31, 2010 | Reply

  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 [...]

    Pingback by Inner and Outer Corners « The Unapologetic Mathematician | January 26, 2011 | Reply

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

    Pingback by Generalized Young Tableaux « The Unapologetic Mathematician | February 2, 2011 | Reply


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Follow

Get every new post delivered to your Inbox.

Join 388 other followers

%d bloggers like this: