Fomin's grid approach to Robinson-Schensted Demo (1.0.2)

This applet demonstrates the 2-D grid approach to the Robinson-Schensted correspondence, develped by Fomin (and extended by T. Roby)

First, enter the size of your permutation in the text field, and press enter. You will see a grid appear in the starting configuration. You can create as many grids as you want this way; at any time, enter a integer in the text field to create a new grid.

Once you have a grid available, select the squares that should be atoms by clicking in them. When you have entered a valid permutation, you will be free to click on the fill  Fomin table button, which will use your selected permutation to calculate a growth from the grid points to partitions. These partitions will be shown at the corners. If you select a square and later decide to unselect it, click in the square again -- clicking on a square toggles between its status as an atom.

Note the checkboxes associated with each grid window -- the first pair allows you to decide whether you want the grid filled gradually so you can watch the local growth rules working, or whether you want to immediately see the partitions computed. The second pair lets you specify whether you want to view the partitions as numerical partitions (e.g. "43321") or as small Young diagrams.

When you have computed the partitions along the top and right sides of the table, you can press the tablueax from upper button to bring up a small window showing the (P,Q) tableau pair which corresponds to the upper boundary of your Fomin table.

If a Fomin table or (P,Q)-tableau widnow gets covered up, press the refresh display button to redraw. When done with a window, simply close the window from the window menu.

Using the Tableau Pair Editor is not hard. Start by clicking in the upper-left corner, near the intersection of the axes. Each time you click, a new box will appear and a highlight will show which box you clicked last. You can click anywhere to add new boxes.

When a box is highlighted, it means that you may enter an integer value to associate with that box. You do this by selecting the text field at the top of the window and entering your integer, whereupon the selected box gets the integer. Selecting a box which already contains a integer allows you to place a new value in that box. But, by clicking on a box again which is already selected, any integer within disappears. This allows you to revert back to an empty box. Finally, to get remove a box entirely, simply shift-click on it. Note the P and Q buttons at the top of the window which allow you to select which member of the tableau pair you want to work in

Of course, the members of a valid tableau pair must have the same shape! So note that the tableau which is not shown will be affected by actions on the visible Tableau in order to keep their shapes the same. For example, if you remove a box in the visible tableau, the corresponding box will be removed from the other tableau. If a box in the visible tableau becomes blank, the corresponding box in the other tableau will become blank also. Finally, if you enter a value in a previously blank box in the visible tableau, if the corresponding box in the other tableau will get the value (arbitrarily) "4."

When you're done working with your tableau pair, press the make Fomin Table to make a Fomin table with your tableau pair providing the upper boundary of the table. This button will be sensitive exactly when the both members of a pair are valid Young tableau objects.

Applet by Dave Goggin