The page can be used in two ways.  First, you can have the page generate a random input parameter.  To do that, enter the size of the parameter in the first textbox.  Optionally, enter the number of pairs in the parameter in the second textbox.  The number of pairs should be less than or equal to half the number chosen for n.

After entering this information, click one of the two buttons directly to the right.  The Run button will display a randomly-generated parameter and the output of the Sp(p,q) Algorithm when applied to that parameter.  Alternatively, the Run Steps button will also generate a random parameter, and will show you the intermediate steps of applying the Sp(p,q) Algorithm to that parameter.

Pressing Enter while in one of the 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.

You can also enter an input parameter in the textbox labelled "parameter".  A parameter is an arrangement of the numbers 1 through n (with n = p + q), where each number is either a singleton or paired with another number.  Any singleton number has a sign associated to it, either + or -.  In addition, each pair has a sign (or orientation) associated with it.  So, for example, this is a parameter:  1+ 3- 2_-5 6+ 4_7.  The _ is used to indicate a pair of numbers.  If a pair has the negative orientation, then the second number in the pair is written with a - sign.  Specifically, this will be a parameter for Sp(p,q) where p = (number of + signs) + (number of pairs), and similarly q = (number of - signs) + (number of pairs).  So, the example above is a parameter for Sp(4,3).

To enter a parameter in the parameter textbox, write the parameter as in the example, with spaces separating the pieces of the parameter.  Then, use the Run or Run Steps buttons to the right of that box, as above.

This page displays the algorithm for computing annihilators and associated varieties for irreducible Harish-Chandra modules for Sp(p,q).  The input to the Sp(p,q) algorithm is a parameter for the irreducible Harish-Chandra module.  For the purposes of this algorithm, if n = p + q, then a parameter takes the form of an arrangement of the numbers 1, ..., n, where each number is either a singleton or paired with another number.  Any singleton number has a sign associated to it, either + or -.  In addition, each pair has a sign (or orientation) associated with it.  So, for example, this is a parameter:  1+ 3- 2_-5 6+ 4_7.  The _ is used to indicate a pair of numbers.  If a pair has the negative orientation, then the second number in the pair is written with a - sign.  We have p = (number of + signs) + (number of pairs), and similarly q = (number of - signs) + (number of pairs).  So, the example above is a parameter for Sp(4,3).

A parameter is determined by its pieces (that is, the number/sign combinations and number/number pairs).  Unlike in a permutation, the order of the pieces doesn't matter.  We will list the pieces in increasing numerical order, where this increasing order disregards the first number of any number pair, and just uses the second number to find its place.  (The example parameter above is in this order.)  The pieces of the parameter are used by the algorithm in this order.

The output of the Sp(p,q) algorithm is a pair of tableaux.  The left tableau is a domino tableau with numbers, just as in the output to the Domino Robinson-Schensted algorithm.  The left tableaux which are output by this algorithm have the additional property that the number of rows of any give length is even.  The right tableau is a sign tableau, which is used by the algorithm, and also can be used to derive the associated variety of the irreducible Harish-Chandra module.  More precisely, the output of the algorithm is an equivalence class of such pairs of tableaux, with a somewhat complicated equivalence relationship.  With this equivalence relationship, the algorithm is a bijection.  The left tableau of the output corresponds to the primitive ideal which is the annihilator of the irreducible Harish-Chandra module associated to the parameter.  The right tableau describes the associated variety of this Harish-Chandra module.

Each of the two tableaux is presented on a grid of 2x2 squares.  This grid is an important feature of the algorithm.  It shows when a tableau shape is special, as well as how to go from special to non-special configurations of the number tableau, when required by the algorithm.  Sp(p,q) is of type C, and so for this algorithm, both tableaux are in the C position with respect to the grid.  In the C position, the top-left corner of the tableau is in the top-left corner of a 2x2 square of the grid.