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Schützenberger Involution

About the Schützenberger Involution

The Schützenberger involution \(S\) is an operation on standard Young tableaux. After applying the involution to a tableau \(T\), the result \(S(T)\) will have the same shape as \(T\), with the property that \(S(S(T)) = T\).

The algorithm makes use of jeu de taquin slides:
  1. Remove the box in the first row and first column of the tableau. Record the value \(m\) of this box.
  2. Perform a jeu de taquin slide to replace this box.
  3. Place a new box in the last position to be emptied in the slide, labelled \(-m\).
  4. Repeat until all of the original boxes of \(T\) have been removed. Note that each slide will move only the original boxes of \(T\), and does not include the newly placed, negative boxes.
After all boxes of the original tableau have been removed in this way, a tableau with the same shape but all negative values is the result. Adding the same positive value (the number of boxes of \(T\) plus one) to each box will result in the standard tableau \(S(T)\).

Using the Applet

Tap on the grid to build a standard tableau \(T\). Each tap on an blue box will add that box to the tableau, and number it in order. The applet will only allow boxes to be added so that the tableau remains standard. Once the tableau is finished, click the button above to begin the involution \(S\). Tap the box in the top left corner, in green, to remove that box and start a jeu de taquin slide to replace it. Once all boxes are removed from the tableau, click the button again to add the appropriate value to all entries, resulting in \(S(T)\). The original tableau entered is recorded at right.

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.