To begin, select a type of tableau for your output, using the radio buttons. Selecting a radio button will change the type used in subsequent runs of the algorithm.
The page can be used in two ways. First, you can have the page generate a random signed permutation. To do that, enter the size of the desired permutation in the first textbox. Then click one of the two buttons directly to the right. The Run button will display a randomly-generated signed permutation and the output of the Robinson-Schensted algorithm when applied to that permutation. Alternatively, the Run Steps button will also generate a random signed permutation, and will show you the intermediate steps of applying the Robinson-Schensted algorithm to that permutation.
Instead, you can enter a permutation in the second textbox. To do that, list the (positive and negative) numbers of the signed permutation, separated by spaces. Then, use the Run or Run Steps buttons to the right of that box, as above.
Pressing Enter while in one of these two textboxes will click the Run button to the right. Pressing a button a second time, or a different button, will clear the previous result as it gives you a new result, as will pressing Enter while in a textbox.
This page displays the Domino Robinson-Schensted algorithm. The Domino Robinson-Schensted algorithm is a relatively straightforward generalization of the Robinson-Schensted algorithm to the hyperoctrahedral group, that is, the group of signed permutations. The input to the Domino Robinson-Schensted algorithm is a "signed permutation", for example, 2 -5 3 6 -4 1. That is, the numbers 1 through n are permuted, and, in addition, each number has a sign (or orientation) associated with it. The output of the Domino Robinson-Schensted algorithm is a pair of domino tableaux with the same shape. That is, instead of a tableau being made of squares, it is made of 2 x 1 shapes (dominos), which can be horizontal or vertical in the tableau. However, unlike the dominos in a game, each domino has only one number in it.
As in the original Robinson-Schensted algorithm, the left tableau is the insertion, or bumping, tableau. The right tableau is the recording tableau. In this algorithm, insertion is either as a horizontal domino in the first row, if the number in the signed permutation is positive, or as a vertical domino in the first column, if the number in the signed permutation is positive. Bumping is also more complicated. A horizontal domino which is bumped entirely moves to the next row below. A vertical domino which is bumped entirely moves to the next column to the right. A domino which is bumped in one square rotates down and to the right on its unbumped square.
After a domino is added to the insertion tableau by this procedure, the resulting change to the shape of the insertion tableau is two squares, adjacent either horizontally or vertically. A domino is then added to the recording tableau to record this change.
You can see an animation of the bumping procedure here: Domino Robinson-Schensted Algorithm Animated.
Domino tableaux are used to answer questions about simple Lie groups of type B, C, or D. To use a domino tableau for a question about a Lie group of a specific type, you need to place the tableau on the grid associated to that type.
The B grid is associated with the group SO(2n + 1). Selecting the B grid will run the Domino Robinson-Schensted algorithm algorithm starting with two tableaux with zeros in their top-left corner. In that way, the resulting tableaux are made of 2n + 1 squares.