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Jeu de Taquin





About Jeu de Taquin

The jeu de taquin (French for 'teasing game') construction is used to transform skew tableau, either into another skew tableau or into a 'straight' standard tableau. It was originally described by Marcel-Paul Schützenberger. A sequence of transformations, called jeu de taquin slides, are applied. The construction defines an equivalence relation, where two skew tableaux are equivalent if they can be obtained from one another by a sequence of slides. Each equivalence class of skew tableaux under this operation will contain exactly one 'straight' standard tableau, called the rectification of the skew tableaux.

There are two types of jeu de taquin slide. Both involve choosing an empty box bordering the skew tableau, so that if the box was added to the tableau, it would remain a skew diagram (or possibly a straight Young diagram). If such a box lies to the north or west of the tableau, it is called an inside corner. If it lies to the east or south, it is an outside corner.

To slide into an inside corner, \(c\):
  1. Find the box \(b_1\) in the tableau bordering \(c\) with the smallest entry. There may only be a single box bordering \(c\); in this case, that box is \(b_1\). Slide \(b_1\) into \(c\).
  2. If \(b_1\), in its former position, had no neighbors to the east or south, the slide is complete.
  3. Otherwise, choose its smallest neighbor \(b_2\), and slide \(b_2\) into the former position of \(b_1\). Repeat steps 2 and 3 until the last box moved had no neighbors to the east or south.
The idea is similar for an outside corner, \(d\):
  1. Find the box \(b_1\) in the tableau bordering \(d\) with the largest entry. There may only be a single box bordering \(d\); in this case, that box is \(b_1\). Slide \(b_1\) into \(d\).
  2. If \(b_1\), in its former position, had no neighbors to the west or north, the slide is complete.
  3. Otherwise, choose its largest neighbor \(b_2\), and slide \(b_2\) into the former position of \(b_1\). Repeat steps 2 and 3 until the last box moved had no neighbors to the west or north.
The two types are slides are inverses of one another. That is, an outside slide can be 'undone' by an inside slide, and vice-versa.

Using the Applet

Tap on the grid to draw the skew diagram. The applet will check that the diagram satisfies the definition of a skew diagram - it must be the difference between two Young diagrams. The applet will allow you to begin with a straight Young diagram, and apply slides to find skew tableaux equivalent to it.

When finished with the diagram, tap the button above the grid to begin numbering. Tap the boxes in order; the first box tapped will be numbered 1, the second will be numbered 2, etc. The applet will indicate in blue which boxes may be tapped at each step to ensure a standard skew tableau.

Once all boxes are numbered, choose a corner to start a slide. The applet will display the valid inside corners in green, with a filled dot, at each step. The outside corners are in pink, with an open dot.

The original skew tableau entered is displayed in the text field to the right.

About this Applet

This applet was created using JavaScript and the Raphael library. If you are unable to see the applet, make sure you have JavaScript enabled in your browser. This applet may not be supported by older browsers.