The Root Game Applet (for vanishing of Schubert calculus)
Summary
The object of the game is to get one token in each square. (The tokens
are the large coloured dots. The small coloured dots are places where
a symbol could be, but isn't.) If it is possible to do this, it proves
that the Schubert structure constant corresponding to the three
permutations entered is non-zero.
How to play
To start a new game, enter three permutations (of size n) into the text
entry fields whose lengths total n*(n-1)/2, and press the "go" button;
or press the random button, to get a random triple of permutations.
Select a move by clicking/dragging on the coloured dots (see below), then
hit the "move" button to execute the move. Repeat the move-making process
until the game is won or lost--in the latter case, the "back" button
will come in handy.
What the mouse buttons do
The easiest way to select a move is to press mouse button 1 or 2 down on a
symbol and drag it (right or up only) to a new square. This will select
a move which will cause that symbol to move to the new square
(this is unique), if such a move exists.
Dragging with mouse button 3 does the same thing, and also immediately
executes the selected move.
Dragging from one coloured dot to another within the same square will cause
a "merge move" to be selected.
Clicking the mouse on the coloured dots also has an effect:
- Mouse button 1: select the symbol and root corresponding to that dot.
- Mouse button 2: select the symbol corresponding to the colour of
that dot.
- Mouse button 3: select the region corresponding to that square.
Finally, if the game is its initial configuration, double clicking with mouse
button 1 or 2 on squares nearest the diagonal will descent cycle.
Known bugs
- The random button doesn't work properly for size>=8.
- Generating random perms in the background slows things down
considerably, and sometimes goes awry.
- Options window may be broken on some browsers.