Testing the invariance of \(F_{8,1}\) and \(F_{8,2}\)

The polynomials \(F_{8,1}\) and \(F_{8,2}\) are constructed to be invariant under the action of \(\operatorname{SL}(6)\) on the second exterior power of the standard representation.

We produce two random elements of \(\operatorname{SL}(6)\) and check that moving points on the Grassmannian by these elements does not change the values of \(F_{8,1}\) and \(F_{8,2}\).

Testing the invariance of \(F_{8,1}\)

First, we load some files and and functions that will be useful.

Macaulay2, version 1.25.11
Type "help" to see useful commands

i1 : load "MukaiModels.m2";

i2 : load "InvariantPolynomialsforMukaiModels.m2";

i3 : matrixFromHyperplaneEquations = (L) -> (
         R:=ring(L_0);
         eqnMatrix:=matrix apply(L, f -> apply(numgens R, i -> coefficient(R_i,f)));
         transpose gens ker eqnMatrix
     );

Next, we compute two random elements of \(\operatorname{SL}(6)\). This was achieved by creating random matrices in \(\operatorname{Mat}_{6 \times 6}\), selecting those that had a nonzero determinant close to 1, and scaling to obtain elements of \(\operatorname{SL}(6)\). We check that the Grassmannian is invariant under these elements.

i4 : g1 = matrix {{4, -5, -2, -8, -1, 7}, {-2, -10, 7, -4, -9, 7}, {-10, -6, 5, 5, 6, 4}, {-3, -4, 9, -2, -3, -6}, {10, -1, -3, -8, -9, 0}, {-1, -3, -3, 5, 3, -1}}

o4 = | 4   -5  -2 -8 -1 7  |
     | -2  -10 7  -4 -9 7  |
     | -10 -6  5  5  6  4  |
     | -3  -4  9  -2 -3 -6 |
     | 10  -1  -3 -8 -9 0  |
     | -1  -3  -3 5  3  -1 |

              6       6
o4 : Matrix ZZ  <-- ZZ

i5 : det(g1)==1

o5 = true

i6 : g2 = matrix {1/3*{-8, 4, -6, 6, 5, -1}, {-4, -8, -1, -4, -6, -2}, {1, -7, -7, -5, -2, -3}, {9, -2, 8, -9, -8, 4}, {-10, -8, -10, -1, 5, -5}, {8, -3, 6, 9, 2, -6}}

o6 = | -8/3 4/3 -2  2  5/3 -1/3 |
     | -4   -8  -1  -4 -6  -2   |
     | 1    -7  -7  -5 -2  -3   |
     | 9    -2  8   -9 -8  4    |
     | -10  -8  -10 -1 5   -5   |
     | 8    -3  6   9  2   -6   |

              6       6
o6 : Matrix QQ  <-- QQ

i7 : det(g2)==1

o7 = true

i8 : rhog1 = 1/1*exteriorPower(2,g1)

o8 = | -50 24  -55 -32 -60 64  -38  35   25  68  42  35  -63  -28 56  |
     | -74 0   -37 -60 -73 30  14   -36  -7  -43 86  22  -43  -67 -46 |
     | -88 60  -8  -50 -74 55  -102 -114 87  21  62  2   -7   -51 -78 |
     | -31 30  -53 -32 -22 76  -15  11   15  22  -3  58  -51  62  27  |
     | -22 3   -62 -8  4   22  -21  -6   60  -6  33  88  -105 38  75  |
     | 22  -75 -34 35  32  -55 48   42   -69 -3  72  52  -66  -22 -24 |
     | 46  8   13  48  32  -8  -26  44   15  64  -70 7   21   56  63  |
     | 102 -64 37  56  76  -68 108  81   -90 -36 -70 7   21   56  63  |
     | 70  -20 23  30  53  -25 30   60   -27 3   -40 4   12   32  36  |
     | 43  -81 21  44  30  -78 57   33   -90 -6  60  -6  -18  -48 -54 |
     | -17 -14 9   12  -49 -34 11   -18  -9  -19 3   26  23   -27 -20 |
     | -4  13  51  -14 -62 23  -15  -57  -6  33  9   31  14   -31 -12 |
     | 24  35  33  -45 -15 40  -24  0    33  -15 14  18  7    -25 -18 |
     | 5   18  39  -17 -26 39  -12  -21  18  9   -3  -14 -27  32  21  |
     | -31 -33 -6  42  -29 -39 21   -30  -36 21  -10 1   3    8   9   |

              15       15
o8 : Matrix QQ   <-- QQ

i9 : rhog2 = 1/1*exteriorPower(2,g2)

o9 = | 80/3  -16/3 -52/3 56/3 32/3 10  68/3  16/3  41/3 -16/3 4     -16/3 11/3  -16/3 -16/3 |
     | 52/3  62/3  -70/3 34/3 22/3 24  11/3  9     47/3 13/3  25/3  -19/3 11/3  -23/3 -17/3 |
     | 36    29    49    24   12   -23 14    -26   -40  -22   14    10    -11   2     14    |
     | -20/3 -10/3 20/3  6    -8   2   19/3  -22/3 8/3  -1    -23/3 14/3  -16/3 5     4     |
     | 80    -23   -66   72   64   41  86    52    56   -22   2     -36   12    -34   -40   |
     | 61    71    -70   36   53   103 10    52    72   22    31    -34   -4    -47   -32   |
     | 104/3 20/3  -88/3 68/3 44/3 22  10/3  20    20/3 35/3  10    -28/3 20/3  -31/3 -20/3 |
     | -48   30    72    -36  -24  -39 -80   -88   -65  -26   0     24    -15   18    40    |
     | -78   -80   14    -51  -33  -43 -15   -51   -55  -27   -35   11    5     22    25    |
     | -92   -10   84    -99  -70  -98 -35   -74   -40  -53   -5    42    0     49    20    |
     | -8/3  0     2     -40  18   -30 -56/3 23/3  -14  -11   56/3  -9    14    -9    -28/3 |
     | 76    -16   -51   -4   -84  15  40    -34   34   46    40    42    18    42    40    |
     | 53    62    -63   49   -78  -33 18    -20   -2   8     18    33    60    57    18    |
     | -11   -10   12    153  -45  126 82    -28   64   54    -86   24    -72   18    40    |
     | 94    20    -78   -82  -75  -84 -60   -1    -50  -47   100   33    90    51    -20   |

              15       15
o9 : Matrix QQ   <-- QQ

i10 : S = QQ[x_12,x_13,x_23,x_14,x_24,x_34,x_15,x_25,x_35,x_45,x_16,x_26,x_36,x_46,x_56];

i11 : Gr26 = ideal {x_12*x_34-x_13*x_24+x_14*x_23,
      x_12*x_35-x_13*x_25+x_15*x_23,
      x_12*x_36-x_13*x_26+x_16*x_23,
      x_12*x_45-x_14*x_25+x_15*x_24,
      x_12*x_46-x_14*x_26+x_16*x_24,
      x_12*x_56-x_15*x_26+x_16*x_25,
      x_13*x_45-x_14*x_35+x_15*x_34,
      x_13*x_46-x_14*x_36+x_16*x_34,
      x_13*x_56-x_15*x_36+x_16*x_35,
      x_14*x_56-x_15*x_46+x_16*x_45,
      x_23*x_45-x_24*x_35+x_25*x_34,
      x_23*x_46-x_24*x_36+x_26*x_34,
      x_23*x_56-x_25*x_36+x_26*x_35,
      x_24*x_56-x_25*x_46+x_26*x_45,
      x_34*x_56-x_35*x_46+x_36*x_45};

o11 : Ideal of S

i12 : F1=map(S,S,transpose rhog1);

o12 : RingMap S <-- S

i13 : F1(Gr26)==Gr26

o13 = true

i14 : F2=map(S,S,transpose rhog2);

o14 : RingMap S <-- S

i15 : F2(Gr26)==Gr26

o15 = true

Next, we choose a random matrix \(M_{8,1} \in \operatorname{Mat}_{8 \times 15}\) and consider the point in \(\operatorname{Gr}(8,15)\) corresponding to its row space. We check that moving this point of the Grassmannian by these group elements does not affect the value of \(F_{8,1}\).

i16 : M81 = matrix {{10, 5, -6, -9, -3, -1, -8, 3, 5, 10, 5, -6, -5, 4, -1}, {-10, 1, 5, -10, 4, 2, -10, -1, 1, 3, -8, 8, 1, -3, -6}, {3, -3, 8, -5, 9, 3, -3, 2, 7, 0, -4, 3, -7, 0, 4}, {4, 4, -4, 2, -8, -5, 9, 8, -3, 0, -9, 5, -5, -10, -3}, {-7, -3, -6, -4, 5, -9, 9, -5, -8, 5, 0, 3, -10, -6, -1}, {-9, -6, 1, 4, 0, 7, 9, -2, 6, 2, -10, 9, 3, 2, -10}, {9, 7, 6, -3, -10, 7, 9, -5, 4, 1, 10, 0, -2, -9, 1}, {7, 9, 10, -4, -3, 7, -8, 7, 3, 4, -5, -8, 1, 0, 2}}

o16 = | 10  5  -6 -9  -3  -1 -8  3  5  10 5   -6 -5  4   -1  |
      | -10 1  5  -10 4   2  -10 -1 1  3  -8  8  1   -3  -6  |
      | 3   -3 8  -5  9   3  -3  2  7  0  -4  3  -7  0   4   |
      | 4   4  -4 2   -8  -5 9   8  -3 0  -9  5  -5  -10 -3  |
      | -7  -3 -6 -4  5   -9 9   -5 -8 5  0   3  -10 -6  -1  |
      | -9  -6 1  4   0   7  9   -2 6  2  -10 9  3   2   -10 |
      | 9   7  6  -3  -10 7  9   -5 4  1  10  0  -2  -9  1   |
      | 7   9  10 -4  -3  7  -8  7  3  4  -5  -8 1   0   2   |

               8       15
o16 : Matrix ZZ  <-- ZZ

i17 : F81(M81)

o17 = -695010642157043109268907317972795169531718595036132744211295820118682770987778372890787840000000000000

o17 : QQ

i18 : F81(transpose(rhog1*transpose(M81)))

o18 = -695010642157043109268907317972795169531718595036132744211295820118682770987778372890787840000000000000

o18 : QQ

i19 : F81(transpose(rhog2*transpose(M81)))

o19 = -695010642157043109268907317972795169531718595036132744211295820118682770987778372890787840000000000000

o19 : QQ

Next, we check that moving the point in \(\operatorname{Gr}(8,15)\) corresponding to the cuspidal curve of genus 8 with octagonal symmetry does not change the value of \(F_{8,1}\).

i20 : R = QQ[x_12,x_13,x_23,x_14,x_24,x_34,x_15,x_25,x_35,x_45,x_16,x_26,x_36,x_46,x_56];

i21 : LCusp8 = {x_23-(5/3)*x_14, x_24-5*x_15, x_25-15*x_16, x_34-20*x_16, x_35-5*x_26, x_45-(5/3)*x_36, x_12-x_56};

i22 : MCusp8 = matrixFromHyperplaneEquations LCusp8

o22 = | 0 1 0   0 0 0  0 0  0 0   0 0 0 0 0 |
      | 0 0 5/3 1 0 0  0 0  0 0   0 0 0 0 0 |
      | 0 0 0   0 5 0  1 0  0 0   0 0 0 0 0 |
      | 0 0 0   0 0 20 0 15 0 0   1 0 0 0 0 |
      | 0 0 0   0 0 0  0 0  5 0   0 1 0 0 0 |
      | 0 0 0   0 0 0  0 0  0 5/3 0 0 1 0 0 |
      | 0 0 0   0 0 0  0 0  0 0   0 0 0 1 0 |
      | 1 0 0   0 0 0  0 0  0 0   0 0 0 0 1 |

               8       15
o22 : Matrix QQ  <-- QQ

i23 : F81(MCusp8)

o23 = 56332941015999638661697319731200000000000000000000000000000

o23 : QQ

i24 : F81(transpose(rhog1*transpose(MCusp8)))

o24 = 56332941015999638661697319731200000000000000000000000000000

o24 : QQ

i25 : F81(transpose(rhog2*transpose(MCusp8)))

o25 = 56332941015999638661697319731200000000000000000000000000000

o25 : QQ

Testing the invariance of \(F_{8,2}\)

We test the invariance of \(F_{8,2}\). We continue to use the elements \(g_1,g_2 \in \operatorname{SL}(6) \) constructed above. We choose a random matrix \(M_{8,2} \in \operatorname{Mat}_{9 \times 15}\) and consider the point in \(\operatorname{Gr}(9,15)\) corresponding to its row space. We also study the point in \(\operatorname{Gr}(9,16)\) corresponding to the tangent developable of genus 8.

i26 : M82 = matrix {{10, 5, -6, -9, -3, -1, -8, 3, 5, 10, 5, -6, -5, 4, -1}, {-10, 1, 5, -10, 4, 2, -10, -1, 1, 3, -8, 8, 1, -3, -6}, {3, -3, 8, -5, 9, 3, -3, 2, 7, 0, -4, 3, -7, 0, 4}, {4, 4, -4, 2, -8, -5, 9, 8, -3, 0, -9, 5, -5, -10, -3}, {-7, -3, -6, -4, 5, -9, 9, -5, -8, 5, 0, 3, -10, -6, -1}, {-9, -6, 1, 4, 0, 7, 9, -2, 6, 2, -10, 9, 3, 2, -10}, {9, 7, 6, -3, -10, 7, 9, -5, 4, 1, 10, 0, -2, -9, 1}, {7, 9, 10, -4, -3, 7, -8, 7, 3, 4, -5, -8, 1, 0, 2},{8, 6, 6, -7, 7, -6, 1, -8, 3, 1, 6, -3, 4, 9, 7}}

o26 = | 10  5  -6 -9  -3  -1 -8  3  5  10 5   -6 -5  4   -1  |
      | -10 1  5  -10 4   2  -10 -1 1  3  -8  8  1   -3  -6  |
      | 3   -3 8  -5  9   3  -3  2  7  0  -4  3  -7  0   4   |
      | 4   4  -4 2   -8  -5 9   8  -3 0  -9  5  -5  -10 -3  |
      | -7  -3 -6 -4  5   -9 9   -5 -8 5  0   3  -10 -6  -1  |
      | -9  -6 1  4   0   7  9   -2 6  2  -10 9  3   2   -10 |
      | 9   7  6  -3  -10 7  9   -5 4  1  10  0  -2  -9  1   |
      | 7   9  10 -4  -3  7  -8  7  3  4  -5  -8 1   0   2   |
      | 8   6  6  -7  7   -6 1   -8 3  1  6   -3 4   9   7   |

               9       15
o26 : Matrix ZZ  <-- ZZ

i27 : F82(M82)

o27 = 15971149687430839382728

o27 : QQ

i28 : F82(transpose(rhog1*transpose(M82)))

o28 = 15971149687430839382728

o28 : QQ

i29 : F82(transpose(rhog2*transpose(M82)))

o29 = 15971149687430839382728

o29 : QQ

i30 : LTD8 = {x_23-(5/3)*x_14, x_24-5*x_15, x_25-15*x_16, x_34-20*x_16, x_35-5*x_26, x_45-(5/3)*x_36};

i31 : MTD8 = matrixFromHyperplaneEquations LTD8;

               9       15
o31 : Matrix QQ  <-- QQ

i32 : F82(MTD8)

        20234375
o32 = - --------
           81

o32 : QQ

i33 : F82(transpose(rhog1*transpose(MTD8)))

        20234375
o33 = - --------
           81

o33 : QQ

i34 : F82(transpose(rhog2*transpose(MTD8)))

        20234375
o34 = - --------
           81

o34 : QQ