-- -*- M2-comint -*- hash: 17730794896671137510

i1 : M = matrix(QQ,{{1,0,1,2},{0,1,2,1}})

o1 = | 1 0 1 2 |
     | 0 1 2 1 |

              2       4
o1 : Matrix QQ  <-- QQ

i2 : X1 = generalizedFlagVariety("A",3,{2})

o2 = a "GKM variety" with an action of a 4-dimensional torus

o2 : GKMVariety

i3 : X2 = generalizedFlagVariety("C",2,{2})

o3 = a "GKM variety" with an action of a 2-dimensional torus

o3 : GKMVariety

i4 : C1 = orbitClosure(X1,M)

o4 = an "equivariant K-class" on a GKM variety 

o4 : KClass

i5 : C2 = orbitClosure(X2,M)

o5 = an "equivariant K-class" on a GKM variety 

o5 : KClass

i6 : peek C1

o6 = KClass{variety => a "GKM variety" with an action of a 4-dimensional torus}
                                                           -1 -1
            KPolynomials => HashTable{{set {0, 1}} => 1 - T  T  T T   }
                                                           0  1  2 3
                                                           -1   -1
                                      {set {0, 2}} => 1 - T  T T  T
                                                           0  1 2  3
                                                           -1     -1
                                      {set {0, 3}} => 1 - T  T T T
                                                           0  1 2 3
                                                           -1 -1
                                      {set {1, 2}} => - T T  T  T  + 1
                                                         0 1  2  3
                                                           -1   -1
                                      {set {1, 3}} => - T T  T T   + 1
                                                         0 1  2 3
                                                             -1 -1
                                      {set {2, 3}} => - T T T  T   + 1
                                                         0 1 2  3

i7 : peek C2

o7 = KClass{variety => a "GKM variety" with an action of a 2-dimensional torus}
                                                            -2 2
            KPolynomials => HashTable{{set {0, 1*}} => 1 - T  T   }
                                                            0  1
                                                           -2 -2
                                      {set {0, 1}} => 1 - T  T
                                                           0  1
                                                           2 2
                                      {set {1*, 0*}} => - T T  + 1
                                                           0 1
                                                          2 -2
                                      {set {1, 0*}} => - T T   + 1
                                                          0 1

i8 : X = generalizedFlagVariety("A",3,{1,2})

o8 = a "GKM variety" with an action of a 4-dimensional torus

o8 : GKMVariety

i9 : Mat = random(QQ^2,QQ^4)

o9 = | 9/2 3/4 7/9  7/10 |
     | 9/4 7/4 7/10 7/3  |

              2       4
o9 : Matrix QQ  <-- QQ

i10 : C = orbitClosure(X,Mat)

o10 = an "equivariant K-class" on a GKM variety 

o10 : KClass

i11 : FM = flagMatroid(Mat,{1,2})

o11 = a "flag matroid" with rank sequence {1, 2} on 4 elements 

o11 : FlagMatroid

i12 : C' = makeKClass(X,FM)

o12 = an "equivariant K-class" on a GKM variety 

o12 : KClass

i13 : C === C'

o13 = true

i14 : R = makeCharacterRing 4

o14 = R

o14 : PolynomialRing

i15 : X1 = generalizedFlagVariety("D",4,{4,4},R)

o15 = a "GKM variety" with an action of a 4-dimensional torus

o15 : GKMVariety

i16 : X2 = generalizedFlagVariety("D",4,{3,3},R)

o16 = a "GKM variety" with an action of a 4-dimensional torus

o16 : GKMVariety

i17 : A = matrix{{1,3,-2,-1/4},{-1,-1,19,-61/4},{0,1,19,-73/4},{2,0,22,-89/4}};

               4       4
o17 : Matrix QQ  <-- QQ

i18 : B = matrix(QQ,{{1,2,3,4},{5,6,7,8},{9,10,11,12},{13,14,15,16}});

               4       4
o18 : Matrix QQ  <-- QQ

i19 : M = A | B

o19 = | 1  3  -2 -1/4  1  2  3  4  |
      | -1 -1 19 -61/4 5  6  7  8  |
      | 0  1  19 -73/4 9  10 11 12 |
      | 2  0  22 -89/4 13 14 15 16 |

o19 : Matri    4       8
x QQ  <-- QQ

i20 : assert(A* transpose(B)  + B *transpose(A) == 0) -- verifying that M is isotropic

i21 : C1 = orbitClosure(X1,M)

o21 = an "equivariant K-class" on a GKM variety 

o21 : KClass

i22 : C2 = orbitClosure(X2,M)

o22 = an "equivariant K-class" on a GKM variety 

o22 : KClass

i23 : peek C1

o23 = KClass{variety => a "GKM variety" with an action of a 4-dimensional torus                       }
                                                                    -1   -1      -1 -1    -2   -2
             KPolynomials => HashTable{{set {0, 1*, 2, 3*}} => 1 - T  T T  T  - T  T   + T  T T  T   }
                                                                    0  1 2  3    0  2     0  1 2  3
                                                                   -1 -1 -1 -1    -2 -2 -2 -2
                                       {set {0, 1, 2, 3}} => 1 - 2T  T  T  T   + T  T  T  T
                                                                   0  1  2  3     0  1  2  3
                                                                    -1     -1    -1 -1    -2     -2
                                       {set {0, 2*, 1*, 3}} => 1 - T  T T T   - T  T   + T  T T T
                                                                    0  1 2 3     0  3     0  1 2 3
                                                                    -1 -1        -1 -1    -2 -2
                                       {set {0, 2*, 1, 3*}} => 1 - T  T  T T  - T  T   + T  T  T T
                                                                    0  1  2 3    0  1     0  1  2 3
                                                                      -1 -1            -2 -2    -1 -1
                                       {set {1*, 0*, 2, 3}} => - T T T  T   + 1 + T T T  T   - T  T
                                                                  0 1 2  3         0 1 2  3     2  3
                                                                    -1 -1            -2 -2      -1 -1
                                       {set {1, 0*, 2, 3*}} => - T T  T  T  + 1 + T T  T  T  - T  T
                                                                  0 1  2  3        0 1  2  3    1  2
                                       {set {2*, 1*, 0*, 3*}} => 0
                                                                    -1   -1          -2   -2    -1 -1
                                       {set {2*, 1, 0*, 3}} => - T T  T T   + 1 + T T  T T   - T  T
                                                                  0 1  2 3         0 1  2 3     1  3

i24 : peek C2 -- since the point corresponding to M lies in X1, C2 is just the empty class i.e. 0

o24 = KClass{KPolynomials => HashTable{{set {0, 1*, 2, 3}} => 0  }             }
                                       {set {0, 1, 2, 3*}} => 0
                                       {set {0, 2*, 1*, 3*}} => 0
                                       {set {0, 2*, 1, 3}} => 0
                                       {set {1*, 0*, 2, 3*}} => 0
                                       {set {1, 0*, 2, 3}} => 0
                                       {set {2*, 1*, 0*, 3}} => 0
                                       {set {2*, 1, 0*, 3*}} => 0
             variety => a "GKM variety" with an action of a 4-dimensional torus

i25 : X = generalizedFlagVariety("A",3,{1,2,3})

o25 = a "GKM variety" with an action of a 4-dimensional torus

o25 : GKMVariety

i26 : Mat = random(QQ^3,QQ^4)

o26 = | 7   6   3/10 10/9 |
      | 3/7 5/4 3/7  10   |
      | 6/7 2/9 5    3/2  |

               3       4
o26 : Matrix QQ  <-- QQ

i27 : time C = orbitClosure(X,Mat)
 -- used 0.488766s (cpu); 0.321972s (thread); 0s (gc)

o27 = an "equivariant K-class" on a GKM variety 

o27 : KClass

i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
 -- used 1.41186s (cpu); 0.837658s (thread); 0s (gc)

o28 = an "equivariant K-class" on a GKM variety 

o28 : KClass

i29 : 
