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 SO(p,q) Odd Sp(n, ℝ) 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 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 0 through n (where 2 * n + 1 = p + q), where each number is either a singleton or paired with another number.  Any singleton number has a sign associated to it.  There are two sets of signs to choose from, so for convenience, signs in the first set are written as + or -, and signs in the second set are written as s or t.  The number 0 always occurs as a singleton, with either a + or a - sign associated to it.  In addition, each pair of numbers has a sign (or orientation) associated with it.  So, for example, this is a parameter:  0- 1+ 3- 2_-5 6s 4_7 8t 9+ 10s 11s.  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 SO(p,q) where p = 2 * (number of + signs) + 2 * (number of pairs) + (number of s signs) + (number of t signs) + (1 if 0 has a + sign), and similarly q = p = 2 * (number of - signs) + 2 * (number of pairs) + (number of s signs) + (number of t signs) + (1 if 0 has a - sign).  So, the example above is a parameter for SO(12, 11).

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.

Warning: this algorithm is still experimental.  If you have questions about it, or other issues, please contact me at devra.johnson at verizon.net.

This algorithm is essentially the same algorithm which is used to compute annihilators and associated varieties for SO(p,q) where p + q is even. I recommend looking at the SO(p,q) Even algorithm first, because it is the more basic version of the algorithm.

This page displays a candidate for the algorithm for computing annihilators and associated varieties for irreducible Harish-Chandra modules for SO(p,q), with p + q odd, and for Sp(n, ℝ).  The input to the SO(p,q) Odd Sp(n, ℝ) algorithm is a parameter for an irreducible Harish-Chandra module for SO(p,q), with p + q odd.  A parameter is an arrangement of the numbers 0 through n (where 2 * n + 1 = p + q), where each number is either a singleton or paired with another number.  Any singleton number has a sign associated to it.  There are two sets of signs to choose from, so for convenience, signs in the first set are written as + or -, and signs in the second set are written as s or t.  The number 0 always occurs as a singleton, with either a + or a - sign associated to it.  In addition, each pair of numbers has a sign (or orientation) associated with it.  So, for example, this is a parameter:  0- 1+ 3- 2_-5 6s 4_7 8t 9+ 10s 11s.  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 SO(p,q) where p = 2 * (number of + signs) + 2 * (number of pairs) + (number of s signs) + (number of t signs) + (1 if 0 has a + sign), and similarly q = p = 2 * (number of - signs) + 2 * (number of pairs) + (number of s signs) + (number of t signs) + (1 if 0 has a - sign).  So, the example above is a parameter for SO(12, 11).

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.

This algorithm is very connected to the duality which is defined in Vogan's paper, "Irreducible characters of semisimple Lie groups IV. Character-multiplicity duality".  Using that duality, a parameter for a representation of SO(p,q), with p + q odd, such as described above, is also a parameter for a dual representation on the group, Sp(n, ℝ). The form of the parameter which we are using is probably not the one which you would choose if you were just looking at one representation.  Please see the page for the SO(p,q) Even algorithm for more discussion fo this.

The output of the SO(p,q) Odd Sp(n, ℝ) algorithm is two pairs of tableaux.  The left tableau of each pair is a domino tableau with numbers, just as in the output to the Domino Robinson-Schensted algorithm.  The right tableau of each pair is a sign tableau, which is used by the algorithm, and also can be used to derive the associated variety of an irreducible Harish-Chandra module.  More precisely, the output of the algorithm is an equivalence class of such pairs of pairs of tableaux, with a complicated description of which such quadruples of tableaux can occur, and also a complicated description of the equivalence relationship.  With this equivalence relationship, the algorithm is conjecturally a bijection.  The left tableau of the first pair corresponds to the primitive idea which is the annihilator of the irreducible Harish-Chandra module associated to the parameter.  The right tableau of the first pair describes the associated variety of this Harish-Chandra module. The left and right tableaux of the second pair correspond in the same way to the annihilator and associated variaty of the dual representation of the original Harish-Chandra module, where duality is defined in Vogan's paper, "Irreducible characters of semisimple Lie groups IV. Character-multiplicity duality".

Each of the four tableaux is presented on a grid of 2 x 2 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.  SO(p,q), with p + q odd, is of type B, and so for this algorithm, the first number tableau is in the B position with respect to its grid.  In the B position, the top-left corner of the tableau is in the bottom-right corner of a 2 x 2 square of the grid.  The group Sp(n, ℝ) is of type C, and so for this algorithm, the right number tableau is 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 2 x 2 square of the grid. 

On the other hand, both sign tableaux are on the D grid.  This is because we are using the same procedure which is used for SO(p,q) with p + q even, and so the sign tableaux need to conform to that algorithm by being on the D grid.  To achieve that for this algorithm, the first sign tableau starts with one vertical domino with the same sign as the 0 in the parameter, and then an (infinite) row with alternating signs to the right of that domino, in the first row of the sign tableau.  The second sign tableau starts with a corresponding (infinite) column of blank dominos in its first column.

Within a pair, the two tableaux (number tableau and sign tableau) have the same shape, except for the extra row or column.  The two number tableau are roughly (but not quite) transposes of each other.  More precisely, you can obtain one from the other by taking a transpose and then moving through some of the open cycles, including the cycle containing the number 1.

You can find out more about the basic ingredients of this algorithm, namely the Domino Robinson-Schensted algorithm and the procedure of moving through cycles, on my website https://devragj.github.io/.