Investigating matrix Schubert varieties -- basic functions for Schubert determinantal ideals

Matrix Schubert varieties were introduced by Fulton [Ful92] in the study of Schubert varieties in the complete flag variety. Their defining ideals are called Schubert determinantal ideals.

[Ful92] William Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math J. 65 (1992): 381-420.
[HPW22] Zachary Hamaker, Oliver Pechenik, and Anna Weigandt, Gröbner geometry of Schubert polynomials through ice, Advances in Mathematics 398 (2022): 108228.
[KM05] Allen Knutson and Ezra Miller, Gröbner geometry of Schubert polynomials, Annals of Mathematics (2005): 1245-1318.
[PSW24] Oliver Pechenik, David Speyer, and Anna Weigandt, Castelnuovo-Mumford regularity of matrix Schubert varieties, Selecta Mathematica New Series 30, 66 (2024).

The general method for creating a Schubert determinantal Ideal is schubertDeterminantalIdeal. The input is a permutation in the form of a list. This package contains functions for investigating the rank matrix (rankTable), the Rothe diagram (rotheDiagram), and the essential cells (essentialSet) of the Rothe diagram as defined by Fulton in [Ful92].

i1 : p = {2,1,6,3,5,4};

i2 : rotheDiagram p

o2 = {(1, 1), (3, 3), (3, 4), (3, 5), (5, 4)}

o2 : List

i3 : essentialSet p

o3 = {(1, 1), (3, 5), (5, 4)}

o3 : List

i4 : rankTable p

o4 = | 0 1 1 1 1 1 |
     | 1 2 2 2 2 2 |
     | 1 2 2 2 2 3 |
     | 1 2 3 3 3 4 |
     | 1 2 3 3 4 5 |
     | 1 2 3 4 5 6 |

              6       6
o4 : Matrix ZZ  <-- ZZ

i5 : netList fultonGens p

     +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
o5 = |z                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
     | 1,1                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
     +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |- z   z   z    + z   z   z    + z   z   z    - z   z   z    - z   z   z    + z   z   z                                                                                                                                                                                                                                                                                                                                                                               |
     |   1,3 2,2 3,1    1,2 2,3 3,1    1,3 2,1 3,2    1,1 2,3 3,2    1,2 2,1 3,3    1,1 2,2 3,3                                                                                                                                                                                                                                                                                                                                                                            |
     +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |- z   z   z    + z   z   z    + z   z   z    - z   z   z    - z   z   z    + z   z   z                                                                                                                                                                                                                                                                                                                                                                               |
     |   1,4 2,2 3,1    1,2 2,4 3,1    1,4 2,1 3,2    1,1 2,4 3,2    1,2 2,1 3,4    1,1 2,2 3,4                                                                                                                                                                                                                                                                                                                                                                            |
     +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |- z   z   z    + z   z   z    + z   z   z    - z   z   z    - z   z   z    + z   z   z                                                                                                                                                                                                                                                                                                                                                                               |
     |   1,4 2,3 3,1    1,3 2,4 3,1    1,4 2,1 3,3    1,1 2,4 3,3    1,3 2,1 3,4    1,1 2,3 3,4                                                                                                                                                                                                                                                                                                                                                                            |
     +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |- z   z   z    + z   z   z    + z   z   z    - z   z   z    - z   z   z    + z   z   z                                                                                                                                                                                                                                                                                                                                                                               |
     |   1,4 2,3 3,2    1,3 2,4 3,2    1,4 2,2 3,3    1,2 2,4 3,3    1,3 2,2 3,4    1,2 2,3 3,4                                                                                                                                                                                                                                                                                                                                                                            |
     +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |- z   z   z    + z   z   z    + z   z   z    - z   z   z    - z   z   z    + z   z   z                                                                                                                                                                                                                                                                                                                                                                               |
     |   1,5 2,2 3,1    1,2 2,5 3,1    1,5 2,1 3,2    1,1 2,5 3,2    1,2 2,1 3,5    1,1 2,2 3,5                                                                                                                                                                                                                                                                                                                                                                            |
     +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |- z   z   z    + z   z   z    + z   z   z    - z   z   z    - z   z   z    + z   z   z                                                                                                                                                                                                                                                                                                                                                                               |
     |   1,5 2,3 3,1    1,3 2,5 3,1    1,5 2,1 3,3    1,1 2,5 3,3    1,3 2,1 3,5    1,1 2,3 3,5                                                                                                                                                                                                                                                                                                                                                                            |
     +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |- z   z   z    + z   z   z    + z   z   z    - z   z   z    - z   z   z    + z   z   z                                                                                                                                                                                                                                                                                                                                                                               |
     |   1,5 2,3 3,2    1,3 2,5 3,2    1,5 2,2 3,3    1,2 2,5 3,3    1,3 2,2 3,5    1,2 2,3 3,5                                                                                                                                                                                                                                                                                                                                                                            |
     +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |- z   z   z    + z   z   z    + z   z   z    - z   z   z    - z   z   z    + z   z   z                                                                                                                                                                                                                                                                                                                                                                               |
     |   1,5 2,4 3,1    1,4 2,5 3,1    1,5 2,1 3,4    1,1 2,5 3,4    1,4 2,1 3,5    1,1 2,4 3,5                                                                                                                                                                                                                                                                                                                                                                            |
     +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |- z   z   z    + z   z   z    + z   z   z    - z   z   z    - z   z   z    + z   z   z                                                                                                                                                                                                                                                                                                                                                                               |
     |   1,5 2,4 3,2    1,4 2,5 3,2    1,5 2,2 3,4    1,2 2,5 3,4    1,4 2,2 3,5    1,2 2,4 3,5                                                                                                                                                                                                                                                                                                                                                                            |
     +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |- z   z   z    + z   z   z    + z   z   z    - z   z   z    - z   z   z    + z   z   z                                                                                                                                                                                                                                                                                                                                                                               |
     |   1,5 2,4 3,3    1,4 2,5 3,3    1,5 2,3 3,4    1,3 2,5 3,4    1,4 2,3 3,5    1,3 2,4 3,5                                                                                                                                                                                                                                                                                                                                                                            |
     +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z   |
     | 1,4 2,3 3,2 4,1    1,3 2,4 3,2 4,1    1,4 2,2 3,3 4,1    1,2 2,4 3,3 4,1    1,3 2,2 3,4 4,1    1,2 2,3 3,4 4,1    1,4 2,3 3,1 4,2    1,3 2,4 3,1 4,2    1,4 2,1 3,3 4,2    1,1 2,4 3,3 4,2    1,3 2,1 3,4 4,2    1,1 2,3 3,4 4,2    1,4 2,2 3,1 4,3    1,2 2,4 3,1 4,3    1,4 2,1 3,2 4,3    1,1 2,4 3,2 4,3    1,2 2,1 3,4 4,3    1,1 2,2 3,4 4,3    1,3 2,2 3,1 4,4    1,2 2,3 3,1 4,4    1,3 2,1 3,2 4,4    1,1 2,3 3,2 4,4    1,2 2,1 3,3 4,4    1,1 2,2 3,3 4,4|
     +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z   |
     | 1,4 2,3 3,2 5,1    1,3 2,4 3,2 5,1    1,4 2,2 3,3 5,1    1,2 2,4 3,3 5,1    1,3 2,2 3,4 5,1    1,2 2,3 3,4 5,1    1,4 2,3 3,1 5,2    1,3 2,4 3,1 5,2    1,4 2,1 3,3 5,2    1,1 2,4 3,3 5,2    1,3 2,1 3,4 5,2    1,1 2,3 3,4 5,2    1,4 2,2 3,1 5,3    1,2 2,4 3,1 5,3    1,4 2,1 3,2 5,3    1,1 2,4 3,2 5,3    1,2 2,1 3,4 5,3    1,1 2,2 3,4 5,3    1,3 2,2 3,1 5,4    1,2 2,3 3,1 5,4    1,3 2,1 3,2 5,4    1,1 2,3 3,2 5,4    1,2 2,1 3,3 5,4    1,1 2,2 3,3 5,4|
     +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z   |
     | 1,4 2,3 4,2 5,1    1,3 2,4 4,2 5,1    1,4 2,2 4,3 5,1    1,2 2,4 4,3 5,1    1,3 2,2 4,4 5,1    1,2 2,3 4,4 5,1    1,4 2,3 4,1 5,2    1,3 2,4 4,1 5,2    1,4 2,1 4,3 5,2    1,1 2,4 4,3 5,2    1,3 2,1 4,4 5,2    1,1 2,3 4,4 5,2    1,4 2,2 4,1 5,3    1,2 2,4 4,1 5,3    1,4 2,1 4,2 5,3    1,1 2,4 4,2 5,3    1,2 2,1 4,4 5,3    1,1 2,2 4,4 5,3    1,3 2,2 4,1 5,4    1,2 2,3 4,1 5,4    1,3 2,1 4,2 5,4    1,1 2,3 4,2 5,4    1,2 2,1 4,3 5,4    1,1 2,2 4,3 5,4|
     +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z   |
     | 1,4 3,3 4,2 5,1    1,3 3,4 4,2 5,1    1,4 3,2 4,3 5,1    1,2 3,4 4,3 5,1    1,3 3,2 4,4 5,1    1,2 3,3 4,4 5,1    1,4 3,3 4,1 5,2    1,3 3,4 4,1 5,2    1,4 3,1 4,3 5,2    1,1 3,4 4,3 5,2    1,3 3,1 4,4 5,2    1,1 3,3 4,4 5,2    1,4 3,2 4,1 5,3    1,2 3,4 4,1 5,3    1,4 3,1 4,2 5,3    1,1 3,4 4,2 5,3    1,2 3,1 4,4 5,3    1,1 3,2 4,4 5,3    1,3 3,2 4,1 5,4    1,2 3,3 4,1 5,4    1,3 3,1 4,2 5,4    1,1 3,3 4,2 5,4    1,2 3,1 4,3 5,4    1,1 3,2 4,3 5,4|
     +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     |z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z    + z   z   z   z    - z   z   z   z    - z   z   z   z    + z   z   z   z   |
     | 2,4 3,3 4,2 5,1    2,3 3,4 4,2 5,1    2,4 3,2 4,3 5,1    2,2 3,4 4,3 5,1    2,3 3,2 4,4 5,1    2,2 3,3 4,4 5,1    2,4 3,3 4,1 5,2    2,3 3,4 4,1 5,2    2,4 3,1 4,3 5,2    2,1 3,4 4,3 5,2    2,3 3,1 4,4 5,2    2,1 3,3 4,4 5,2    2,4 3,2 4,1 5,3    2,2 3,4 4,1 5,3    2,4 3,1 4,2 5,3    2,1 3,4 4,2 5,3    2,2 3,1 4,4 5,3    2,1 3,2 4,4 5,3    2,3 3,2 4,1 5,4    2,2 3,3 4,1 5,4    2,3 3,1 4,2 5,4    2,1 3,3 4,2 5,4    2,2 3,1 4,3 5,4    2,1 3,2 4,3 5,4|
     +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+

The default presentation given by schubertDeterminantalIdeal is given by the Fulton generators of the ideal. In order to access a minimal generating set, use trim.

i6 : I = schubertDeterminantalIdeal p;

o6 : Ideal of QQ[z   ..z   ]
                  1,1   6,6

i7 : numgens I

o7 = 16

i8 : numgens (trim I)

o8 = 14

After creating a Schubert determinantal ideal, the permutation associated to it is stored in its cache table under the key ASM. One can also access the permutation matrix directly using getASM.

i9 : peek I.cache

o9 = CacheTable{ASM => | 0 1 0 0 0 0 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               }
                       | 1 0 0 0 0 0 |
                       | 0 0 0 0 0 1 |
                       | 0 0 1 0 0 0 |
                       | 0 0 0 0 1 0 |
                       | 0 0 0 1 0 0 |
                module => image | z_(1,1) -z_(1,3)z_(2,2)z_(3,1)+z_(1,2)z_(2,3)z_(3,1)+z_(1,3)z_(2,1)z_(3,2)-z_(1,1)z_(2,3)z_(3,2)-z_(1,2)z_(2,1)z_(3,3)+z_(1,1)z_(2,2)z_(3,3) -z_(1,4)z_(2,2)z_(3,1)+z_(1,2)z_(2,4)z_(3,1)+z_(1,4)z_(2,1)z_(3,2)-z_(1,1)z_(2,4)z_(3,2)-z_(1,2)z_(2,1)z_(3,4)+z_(1,1)z_(2,2)z_(3,4) -z_(1,4)z_(2,3)z_(3,1)+z_(1,3)z_(2,4)z_(3,1)+z_(1,4)z_(2,1)z_(3,3)-z_(1,1)z_(2,4)z_(3,3)-z_(1,3)z_(2,1)z_(3,4)+z_(1,1)z_(2,3)z_(3,4) -z_(1,4)z_(2,3)z_(3,2)+z_(1,3)z_(2,4)z_(3,2)+z_(1,4)z_(2,2)z_(3,3)-z_(1,2)z_(2,4)z_(3,3)-z_(1,3)z_(2,2)z_(3,4)+z_(1,2)z_(2,3)z_(3,4) -z_(1,5)z_(2,2)z_(3,1)+z_(1,2)z_(2,5)z_(3,1)+z_(1,5)z_(2,1)z_(3,2)-z_(1,1)z_(2,5)z_(3,2)-z_(1,2)z_(2,1)z_(3,5)+z_(1,1)z_(2,2)z_(3,5) -z_(1,5)z_(2,3)z_(3,1)+z_(1,3)z_(2,5)z_(3,1)+z_(1,5)z_(2,1)z_(3,3)-z_(1,1)z_(2,5)z_(3,3)-z_(1,3)z_(2,1)z_(3,5)+z_(1,1)z_(2,3)z_(3,5) -z_(1,5)z_(2,3)z_(3,2)+z_(1,3)z_(2,5)z_(3,2)+z_(1,5)z_(2,2)z_(3,3)-z_(1,2)z_(2,5)z_(3,3)-z_(1,3)z_(2,2)z_(3,5)+z_(1,2)z_(2,3)z_(3,5) -z_(1,5)z_(2,4)z_(3,1)+z_(1,4)z_(2,5)z_(3,1)+z_(1,5)z_(2,1)z_(3,4)-z_(1,1)z_(2,5)z_(3,4)-z_(1,4)z_(2,1)z_(3,5)+z_(1,1)z_(2,4)z_(3,5) -z_(1,5)z_(2,4)z_(3,2)+z_(1,4)z_(2,5)z_(3,2)+z_(1,5)z_(2,2)z_(3,4)-z_(1,2)z_(2,5)z_(3,4)-z_(1,4)z_(2,2)z_(3,5)+z_(1,2)z_(2,4)z_(3,5) -z_(1,5)z_(2,4)z_(3,3)+z_(1,4)z_(2,5)z_(3,3)+z_(1,5)z_(2,3)z_(3,4)-z_(1,3)z_(2,5)z_(3,4)-z_(1,4)z_(2,3)z_(3,5)+z_(1,3)z_(2,4)z_(3,5) z_(1,4)z_(2,3)z_(3,2)z_(4,1)-z_(1,3)z_(2,4)z_(3,2)z_(4,1)-z_(1,4)z_(2,2)z_(3,3)z_(4,1)+z_(1,2)z_(2,4)z_(3,3)z_(4,1)+z_(1,3)z_(2,2)z_(3,4)z_(4,1)-z_(1,2)z_(2,3)z_(3,4)z_(4,1)-z_(1,4)z_(2,3)z_(3,1)z_(4,2)+z_(1,3)z_(2,4)z_(3,1)z_(4,2)+z_(1,4)z_(2,1)z_(3,3)z_(4,2)-z_(1,1)z_(2,4)z_(3,3)z_(4,2)-z_(1,3)z_(2,1)z_(3,4)z_(4,2)+z_(1,1)z_(2,3)z_(3,4)z_(4,2)+z_(1,4)z_(2,2)z_(3,1)z_(4,3)-z_(1,2)z_(2,4)z_(3,1)z_(4,3)-z_(1,4)z_(2,1)z_(3,2)z_(4,3)+z_(1,1)z_(2,4)z_(3,2)z_(4,3)+z_(1,2)z_(2,1)z_(3,4)z_(4,3)-z_(1,1)z_(2,2)z_(3,4)z_(4,3)-z_(1,3)z_(2,2)z_(3,1)z_(4,4)+z_(1,2)z_(2,3)z_(3,1)z_(4,4)+z_(1,3)z_(2,1)z_(3,2)z_(4,4)-z_(1,1)z_(2,3)z_(3,2)z_(4,4)-z_(1,2)z_(2,1)z_(3,3)z_(4,4)+z_(1,1)z_(2,2)z_(3,3)z_(4,4) z_(1,4)z_(2,3)z_(3,2)z_(5,1)-z_(1,3)z_(2,4)z_(3,2)z_(5,1)-z_(1,4)z_(2,2)z_(3,3)z_(5,1)+z_(1,2)z_(2,4)z_(3,3)z_(5,1)+z_(1,3)z_(2,2)z_(3,4)z_(5,1)-z_(1,2)z_(2,3)z_(3,4)z_(5,1)-z_(1,4)z_(2,3)z_(3,1)z_(5,2)+z_(1,3)z_(2,4)z_(3,1)z_(5,2)+z_(1,4)z_(2,1)z_(3,3)z_(5,2)-z_(1,1)z_(2,4)z_(3,3)z_(5,2)-z_(1,3)z_(2,1)z_(3,4)z_(5,2)+z_(1,1)z_(2,3)z_(3,4)z_(5,2)+z_(1,4)z_(2,2)z_(3,1)z_(5,3)-z_(1,2)z_(2,4)z_(3,1)z_(5,3)-z_(1,4)z_(2,1)z_(3,2)z_(5,3)+z_(1,1)z_(2,4)z_(3,2)z_(5,3)+z_(1,2)z_(2,1)z_(3,4)z_(5,3)-z_(1,1)z_(2,2)z_(3,4)z_(5,3)-z_(1,3)z_(2,2)z_(3,1)z_(5,4)+z_(1,2)z_(2,3)z_(3,1)z_(5,4)+z_(1,3)z_(2,1)z_(3,2)z_(5,4)-z_(1,1)z_(2,3)z_(3,2)z_(5,4)-z_(1,2)z_(2,1)z_(3,3)z_(5,4)+z_(1,1)z_(2,2)z_(3,3)z_(5,4) z_(1,4)z_(2,3)z_(4,2)z_(5,1)-z_(1,3)z_(2,4)z_(4,2)z_(5,1)-z_(1,4)z_(2,2)z_(4,3)z_(5,1)+z_(1,2)z_(2,4)z_(4,3)z_(5,1)+z_(1,3)z_(2,2)z_(4,4)z_(5,1)-z_(1,2)z_(2,3)z_(4,4)z_(5,1)-z_(1,4)z_(2,3)z_(4,1)z_(5,2)+z_(1,3)z_(2,4)z_(4,1)z_(5,2)+z_(1,4)z_(2,1)z_(4,3)z_(5,2)-z_(1,1)z_(2,4)z_(4,3)z_(5,2)-z_(1,3)z_(2,1)z_(4,4)z_(5,2)+z_(1,1)z_(2,3)z_(4,4)z_(5,2)+z_(1,4)z_(2,2)z_(4,1)z_(5,3)-z_(1,2)z_(2,4)z_(4,1)z_(5,3)-z_(1,4)z_(2,1)z_(4,2)z_(5,3)+z_(1,1)z_(2,4)z_(4,2)z_(5,3)+z_(1,2)z_(2,1)z_(4,4)z_(5,3)-z_(1,1)z_(2,2)z_(4,4)z_(5,3)-z_(1,3)z_(2,2)z_(4,1)z_(5,4)+z_(1,2)z_(2,3)z_(4,1)z_(5,4)+z_(1,3)z_(2,1)z_(4,2)z_(5,4)-z_(1,1)z_(2,3)z_(4,2)z_(5,4)-z_(1,2)z_(2,1)z_(4,3)z_(5,4)+z_(1,1)z_(2,2)z_(4,3)z_(5,4) z_(1,4)z_(3,3)z_(4,2)z_(5,1)-z_(1,3)z_(3,4)z_(4,2)z_(5,1)-z_(1,4)z_(3,2)z_(4,3)z_(5,1)+z_(1,2)z_(3,4)z_(4,3)z_(5,1)+z_(1,3)z_(3,2)z_(4,4)z_(5,1)-z_(1,2)z_(3,3)z_(4,4)z_(5,1)-z_(1,4)z_(3,3)z_(4,1)z_(5,2)+z_(1,3)z_(3,4)z_(4,1)z_(5,2)+z_(1,4)z_(3,1)z_(4,3)z_(5,2)-z_(1,1)z_(3,4)z_(4,3)z_(5,2)-z_(1,3)z_(3,1)z_(4,4)z_(5,2)+z_(1,1)z_(3,3)z_(4,4)z_(5,2)+z_(1,4)z_(3,2)z_(4,1)z_(5,3)-z_(1,2)z_(3,4)z_(4,1)z_(5,3)-z_(1,4)z_(3,1)z_(4,2)z_(5,3)+z_(1,1)z_(3,4)z_(4,2)z_(5,3)+z_(1,2)z_(3,1)z_(4,4)z_(5,3)-z_(1,1)z_(3,2)z_(4,4)z_(5,3)-z_(1,3)z_(3,2)z_(4,1)z_(5,4)+z_(1,2)z_(3,3)z_(4,1)z_(5,4)+z_(1,3)z_(3,1)z_(4,2)z_(5,4)-z_(1,1)z_(3,3)z_(4,2)z_(5,4)-z_(1,2)z_(3,1)z_(4,3)z_(5,4)+z_(1,1)z_(3,2)z_(4,3)z_(5,4) z_(2,4)z_(3,3)z_(4,2)z_(5,1)-z_(2,3)z_(3,4)z_(4,2)z_(5,1)-z_(2,4)z_(3,2)z_(4,3)z_(5,1)+z_(2,2)z_(3,4)z_(4,3)z_(5,1)+z_(2,3)z_(3,2)z_(4,4)z_(5,1)-z_(2,2)z_(3,3)z_(4,4)z_(5,1)-z_(2,4)z_(3,3)z_(4,1)z_(5,2)+z_(2,3)z_(3,4)z_(4,1)z_(5,2)+z_(2,4)z_(3,1)z_(4,3)z_(5,2)-z_(2,1)z_(3,4)z_(4,3)z_(5,2)-z_(2,3)z_(3,1)z_(4,4)z_(5,2)+z_(2,1)z_(3,3)z_(4,4)z_(5,2)+z_(2,4)z_(3,2)z_(4,1)z_(5,3)-z_(2,2)z_(3,4)z_(4,1)z_(5,3)-z_(2,4)z_(3,1)z_(4,2)z_(5,3)+z_(2,1)z_(3,4)z_(4,2)z_(5,3)+z_(2,2)z_(3,1)z_(4,4)z_(5,3)-z_(2,1)z_(3,2)z_(4,4)z_(5,3)-z_(2,3)z_(3,2)z_(4,1)z_(5,4)+z_(2,2)z_(3,3)z_(4,1)z_(5,4)+z_(2,3)z_(3,1)z_(4,2)z_(5,4)-z_(2,1)z_(3,3)z_(4,2)z_(5,4)-z_(2,2)z_(3,1)z_(4,3)z_(5,4)+z_(2,1)z_(3,2)z_(4,3)z_(5,4) |

i10 : getASM I

o10 = | 0 1 0 0 0 0 |
      | 1 0 0 0 0 0 |
      | 0 0 0 0 0 1 |
      | 0 0 1 0 0 0 |
      | 0 0 0 0 1 0 |
      | 0 0 0 1 0 0 |

               6       6
o10 : Matrix ZZ  <-- ZZ

This package also contains methods for investigating antidiagonal initial ideals (antiDiagInit) of Schubert determinantal ideals and their associated Stanley-Reisner complexes, which are a kind of subword complex (subwordComplex). Subword complexes were introduced in [KM05] in the study of antidiagonal initial ideal of Schubert determinantal ideals and their relation to Schubert polynomials.

i11 : antiDiagInit p

o11 = monomialIdeal (z   , z   z   z   , z   z   z   , z   z   z   ,
                      1,1   1,3 2,2 3,1   1,4 2,2 3,1   1,5 2,2 3,1 
      -----------------------------------------------------------------------
      z   z   z   , z   z   z   , z   z   z   , z   z   z   , z   z   z   ,
       1,4 2,3 3,1   1,5 2,3 3,1   1,5 2,4 3,1   1,4 2,3 3,2   1,5 2,3 3,2 
      -----------------------------------------------------------------------
      z   z   z   , z   z   z   , z   z   z   z   , z   z   z   z   ,
       1,5 2,4 3,2   1,5 2,4 3,3   1,4 2,3 4,2 5,1   1,4 3,3 4,2 5,1 
      -----------------------------------------------------------------------
      z   z   z   z   )
       2,4 3,3 4,2 5,1

o11 : MonomialIdeal of QQ[z   ..z   ]
                           1,1   6,6

i12 : subwordComplex p

o12 = simplicialComplex | z_(1,2)z_(1,6)z_(2,1)z_(2,2)z_(2,3)z_(2,5)z_(2,6)z_(3,1)z_(3,2)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,6)z_(2,1)z_(2,3)z_(2,5)z_(2,6)z_(3,1)z_(3,2)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,5)z_(1,6)z_(2,1)z_(2,5)z_(2,6)z_(3,1)z_(3,2)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,6)z_(2,1)z_(2,2)z_(2,3)z_(2,5)z_(2,6)z_(3,2)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,5)z_(1,6)z_(2,1)z_(2,2)z_(2,5)z_(2,6)z_(3,2)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,5)z_(1,6)z_(2,1)z_(2,2)z_(2,3)z_(2,5)z_(2,6)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,6)z_(2,1)z_(2,2)z_(2,3)z_(2,4)z_(2,5)z_(2,6)z_(3,1)z_(3,2)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,6)z_(2,1)z_(2,3)z_(2,4)z_(2,5)z_(2,6)z_(3,1)z_(3,2)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,4)z_(1,6)z_(2,1)z_(2,4)z_(2,5)z_(2,6)z_(3,1)z_(3,2)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,4)z_(1,5)z_(1,6)z_(2,1)z_(2,5)z_(2,6)z_(3,1)z_(3,2)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,6)z_(2,1)z_(2,2)z_(2,3)z_(2,4)z_(2,5)z_(2,6)z_(3,2)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,4)z_(1,6)z_(2,1)z_(2,2)z_(2,4)z_(2,5)z_(2,6)z_(3,2)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,4)z_(1,5)z_(1,6)z_(2,1)z_(2,2)z_(2,5)z_(2,6)z_(3,2)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,5)z_(1,6)z_(2,1)z_(2,2)z_(2,3)z_(2,4)z_(2,5)z_(2,6)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,4)z_(1,5)z_(1,6)z_(2,1)z_(2,2)z_(2,4)z_(2,5)z_(2,6)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,6)z_(2,1)z_(2,2)z_(2,3)z_(2,4)z_(2,5)z_(2,6)z_(3,1)z_(3,2)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,6)z_(2,1)z_(2,3)z_(2,4)z_(2,5)z_(2,6)z_(3,1)z_(3,2)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,4)z_(1,6)z_(2,1)z_(2,4)z_(2,5)z_(2,6)z_(3,1)z_(3,2)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,4)z_(1,5)z_(1,6)z_(2,1)z_(2,5)z_(2,6)z_(3,1)z_(3,2)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,6)z_(2,1)z_(2,2)z_(2,3)z_(2,4)z_(2,5)z_(2,6)z_(3,2)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,4)z_(1,6)z_(2,1)z_(2,2)z_(2,4)z_(2,5)z_(2,6)z_(3,2)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,4)z_(1,5)z_(1,6)z_(2,1)z_(2,2)z_(2,5)z_(2,6)z_(3,2)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,4)z_(1,6)z_(2,1)z_(2,2)z_(2,3)z_(2,4)z_(2,5)z_(2,6)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,4)z_(1,5)z_(1,6)z_(2,1)z_(2,2)z_(2,3)z_(2,5)z_(2,6)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,4)z_(1,5)z_(1,6)z_(2,1)z_(2,2)z_(2,3)z_(2,4)z_(2,5)z_(2,6)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,1)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,6)z_(2,1)z_(2,2)z_(2,3)z_(2,4)z_(2,5)z_(2,6)z_(3,1)z_(3,2)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,6)z_(2,1)z_(2,3)z_(2,4)z_(2,5)z_(2,6)z_(3,1)z_(3,2)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,4)z_(1,6)z_(2,1)z_(2,4)z_(2,5)z_(2,6)z_(3,1)z_(3,2)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,4)z_(1,5)z_(1,6)z_(2,1)z_(2,5)z_(2,6)z_(3,1)z_(3,2)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,6)z_(2,1)z_(2,2)z_(2,3)z_(2,4)z_(2,5)z_(2,6)z_(3,2)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,4)z_(1,6)z_(2,1)z_(2,2)z_(2,4)z_(2,5)z_(2,6)z_(3,2)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,4)z_(1,5)z_(1,6)z_(2,1)z_(2,2)z_(2,5)z_(2,6)z_(3,2)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,4)z_(1,6)z_(2,1)z_(2,2)z_(2,3)z_(2,4)z_(2,5)z_(2,6)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,4)z_(1,5)z_(1,6)z_(2,1)z_(2,2)z_(2,3)z_(2,5)z_(2,6)z_(3,3)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) z_(1,2)z_(1,3)z_(1,4)z_(1,5)z_(1,6)z_(2,1)z_(2,2)z_(2,3)z_(2,4)z_(2,5)z_(2,6)z_(3,4)z_(3,5)z_(3,6)z_(4,1)z_(4,2)z_(4,3)z_(4,4)z_(4,5)z_(4,6)z_(5,2)z_(5,3)z_(5,4)z_(5,5)z_(5,6)z_(6,1)z_(6,2)z_(6,3)z_(6,4)z_(6,5)z_(6,6) |

o12 : SimplicialComplex

Given a list of permutations, this package also contains functions for intersecting (schubertIntersect) and adding (schubertAdd) the Schubert determinantal ideals associated to the list of permutations.

i13 : L = {{3,1,5,4,2},{2,5,3,4,1}} -- a list of 2 permutations

o13 = {{3, 1, 5, 4, 2}, {2, 5, 3, 4, 1}}

o13 : List

i14 : schubertAdd L

o14 = ideal (z   , z   , - z   z    + z   z   , - z   z    + z   z   , -
              1,1   1,2     1,2 2,1    1,1 2,2     1,3 2,1    1,1 2,3   
      -----------------------------------------------------------------------
      z   z    + z   z   , - z   z    + z   z   , - z   z    + z   z   , -
       1,3 2,2    1,2 2,3     1,4 2,1    1,1 2,4     1,4 2,2    1,2 2,4   
      -----------------------------------------------------------------------
      z   z    + z   z   , z   , z   , z   )
       1,4 2,3    1,3 2,4   2,1   3,1   4,1

o14 : Ideal of QQ[z   ..z   ]
                   1,1   5,5

i15 : schubertIntersect L

                                                                      
o15 = ideal (z   , z   z    - z   z   , z   z    - z   z   , z   z   ,
              1,1   3,2 4,1    3,1 4,2   2,2 4,1    2,1 4,2   1,2 4,1 
      -----------------------------------------------------------------------
                                                                            
      z   z    - z   z   , z   z   , z   z   , z   z   z    - z   z   z    -
       2,2 3,1    2,1 3,2   1,2 3,1   1,2 2,1   1,4 2,3 3,2    1,3 2,4 3,2  
      -----------------------------------------------------------------------
                                                                             
      z   z   z    + z   z   z    + z   z   z    - z   z   z   , z   z   z   
       1,4 2,2 3,3    1,2 2,4 3,3    1,3 2,2 3,4    1,2 2,3 3,4   1,4 2,3 3,1
      -----------------------------------------------------------------------
                                                                  
      - z   z   z    - z   z   z    + z   z   z   , z   z   z    -
         1,3 2,4 3,1    1,4 2,1 3,3    1,3 2,1 3,4   1,2 1,4 2,3  
      -----------------------------------------------------------------------
                                    2                        2
      z   z   z   , z   z   z    - z   z   , z   z   z    - z   z   )
       1,2 1,3 2,4   1,2 1,4 2,2    1,2 2,4   1,2 1,3 2,2    1,2 2,3

o15 : Ideal of QQ[z   ..z   ]
                   1,1   5,5

Finally, this package contains functions for investigating homological invariants of matrix Schubert varieties efficiently through combinatorial algorithms produced in [PSW24] via schubertRegularity schubertCodim.

i16 : time schubertRegularity p
 -- used 0.000996207s (cpu); 0.00037575s (thread); 0s (gc)

o16 = 5

i17 : time regularity comodule I
 -- used 0.0249182s (cpu); 0.0249987s (thread); 0s (gc)

o17 = 5

Functions for investigating matrix Schubert varieties

antiDiagInit(List) -- compute the (unique) antidiagonal initial ideal of an ASM ideal
rankTable(List) -- compute a table of rank conditions that determines the corresponding ASM or matrix Schubert variety
rotheDiagram(List) -- find the Rothe diagram of a partial alternating sign matrix
augmentedRotheDiagram(List) -- find the Rothe diagram and rank table for a partial ASM or permutation
essentialSet(List) -- compute the essential set in the Rothe Diagram for a partial ASM or a permutation.
augmentedEssentialSet(List) -- find the essential set and rank conditions for a partial ASM or a permutation
schubertDeterminantalIdeal(List) -- compute an alternating sign matrix ideal (for example, a Schubert determinantal ideal)
fultonGens(List) -- compute the Fulton generators of an ASM ideal (for example, a Schubert determinantal ideal)
subwordComplex(List) -- to find the subword complex associated to w
schubertIntersect(List) -- compute the intersection of ASM ideals
schubertAdd(List) -- compute the sum of ASM ideals
schubertRegularity(List) -- compute the Castelnuovo-Mumford regularity of the quotient by an ASM ideal
schubertCodim(List) -- compute the codimension (i.e., height) of a Schubert determinantal ideal or ASM ideal

The source of this document is in MatrixSchubert/MatrixSchubertConstructionsDOC.m2:157:0.