Testing the invariance of \(F_{7,1}\) and \(F_{7,2}\)

The polynomials \(F_{7,1}\) and \(F_{7,2}\) are constructed to be invariant under the action of \(\operatorname{Spin}(10)\) on the half-spin representation \(S^{+}\). We check this below as follows.

Every element of \(\operatorname{SO}(2n)\) may be written as a product of an even number of reflections, and elements of \(\operatorname{Spin}(2n)\) may be expressed as certain words of even length in the Clifford algebra. See for instance Fulton and Harris, Representation Theory, Prop. 20.28 and Exercise 20.32.

We test that \(F_{7,1}\) and \(F_{7,2}\) are invariant under some products of this form. First, we test this for a pair of vectors that are simple enough that the action on \(S^{+}\) can be checked by hand. Then, we test this for some pairs of random vectors.

Testing the invariance of \(F_{7,1}\)

Recall that for negative indices, the standard basis vector \(e_{-i}\) acts on the Clifford algebra as contracting by \(e_i\), and for positive indices, the standard basis vector \(e_{i}\) acts as wedging by \(e_i\).

Thus, the product \( (e_{-2}+e_{2}) \cdot (e_{-1} + e_{1}) \) acts on the basis of \(S^{+}\) as follows. \[ \begin{array}{lllll} (e_{-2}+e_{2}) \cdot (e_{-1} +e_{1}) \cdot 1& = & (e_{-2}+e_{2}) \cdot e_1 & = & e_2 \wedge e_1 \\ (e_{-2}+e_{2}) \cdot (e_{-1} +e_{1}) \cdot e_1 \wedge e_2 & = & (e_{-2}+e_{2}) \cdot e_2 & = & 1 \\ (e_{-2}+e_{2}) \cdot (e_{-1} +e_{1}) \cdot e_1 \wedge e_3 & = & (e_{-2}+e_{2}) \cdot e_3 & = & e_2 \wedge e_31\\ (e_{-2}+e_{2}) \cdot (e_{-1} +e_{1}) \cdot e_1 \wedge e_4 & = & (e_{-2}+e_{2}) \cdot e_4 & = & e_2 \wedge e_4\\ (e_{-2}+e_{2}) \cdot (e_{-1} +e_{1}) \cdot e_1 \wedge e_5 & = & (e_{-2}+e_{2}) \cdot e_5 & = & e_2 \wedge e_5\\ (e_{-2}+e_{2}) \cdot (e_{-1} +e_{1}) \cdot e_2 \wedge e_3 & = & (e_{-2}+e_{2}) \cdot e_1 \wedge e_2 \wedge e_3 & = & -e_1 \wedge e_3\\ (e_{-2}+e_{2}) \cdot (e_{-1} +e_{1}) \cdot e_2 \wedge e_4 & = & (e_{-2}+e_{2}) \cdot e_1 \wedge e_2 \wedge e_4 & = & -e_1 \wedge e_4\\ (e_{-2}+e_{2}) \cdot (e_{-1} +e_{1}) \cdot e_2 \wedge e_5 & = & (e_{-2}+e_{2}) \cdot e_1 \wedge e_2 \wedge e_5 & = & -e_1 \wedge e_5\\ (e_{-2}+e_{2}) \cdot (e_{-1} +e_{1}) \cdot e_3 \wedge e_4 & = & (e_{-2}+e_{2}) \cdot e_1 \wedge e_3 \wedge e_4 & = & e_2 \wedge e_1 \wedge e_3 \wedge e_4\\ (e_{-2}+e_{2}) \cdot (e_{-1} +e_{1}) \cdot e_3 \wedge e_5 & = & (e_{-2}+e_{2}) \cdot e_1 \wedge e_3 \wedge e_5 & = & e_2 \wedge e_1 \wedge e_3 \wedge e_5\\ (e_{-2}+e_{2}) \cdot (e_{-1} +e_{1}) \cdot e_4 \wedge e_5 & = & (e_{-2}+e_{2}) \cdot e_1 \wedge e_4 \wedge e_5 & = & e_2 \wedge e_1 \wedge e_4\wedge e_5\\ (e_{-2}+e_{2}) \cdot (e_{-1} +e_{1}) \cdot e_1 \wedge e_2 \wedge e_3 \wedge e_4 & = & (e_{-2}+e_{2}) \cdot e_2 \wedge e_3 \wedge e_4 & = & e_3 \wedge e_4\\ (e_{-2}+e_{2}) \cdot (e_{-1} +e_{1}) \cdot e_1 \wedge e_2 \wedge e_3 \wedge e_5 & = & (e_{-2}+e_{2}) \cdot e_2 \wedge e_3 \wedge e_5 & = & e_3 \wedge e_5\\ (e_{-2}+e_{2}) \cdot (e_{-1} +e_{1}) \cdot e_1 \wedge e_2 \wedge e_4 \wedge e_5 & = & (e_{-2}+e_{2}) \cdot e_2 \wedge e_4 \wedge e_5 & = & e_4 \wedge e_5\\ (e_{-2}+e_{2}) \cdot (e_{-1} +e_{1}) \cdot e_1 \wedge e_3 \wedge e_4 \wedge e_5 & = & (e_{-2}+e_{2}) \cdot e_3 \wedge e_4 \wedge e_5 & = & e_2 \wedge e_3 \wedge e_4 \wedge e_5\\ (e_{-2}+e_{2}) \cdot (e_{-1} +e_{1}) \cdot e_2 \wedge e_3 \wedge e_4 \wedge e_5 & = & (e_{-2}+e_{2}) \cdot e_1 \wedge e_2 \wedge e_3 \wedge e_4 \wedge e_5 & = & -e_1 \wedge e_3 \wedge e_4 \wedge e_5 \end{array} \] This matches the matrix \(g_1\) computed in the Macaulay2 session below. 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 "Invariant_Polynomials_for_Mukai_Models.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
     );

i4 : Q = v -> sum apply(5, i -> v_i*v_(i+5));

i5 : FullContractAndWedgeList=join(apply(5, i -> contractByemk(i+1,wedgeBasis,wedgeBasis,QQ)),apply(5, i -> wedgeByek(i+1,wedgeBasis,wedgeBasis,QQ)));

i6 : Splusindices=apply(2^4, i -> i);

We enter \(e_{-2}+e_{2} \) and \(e_{-1} +e_{1}\) as vectors, and compute the element \(g_1 \in \operatorname{Spin}(10)\).

i7 : ReflectionVectorList1 = {{0,1,0,0,0, 0,1,0,0,0},{1,0,0,0,0,1,0,0,0,0}};

i8 : all(ReflectionVectorList1, v -> Q(v)==1)

o8 = true

i9 : g1 = actionOnSplus(ReflectionVectorList1,FullContractAndWedgeList,Splusindices)

o9 = | 0  1 0 0 0 0  0  0  0  0  0  0 0 0 0 0  |
     | -1 0 0 0 0 0  0  0  0  0  0  0 0 0 0 0  |
     | 0  0 0 0 0 -1 0  0  0  0  0  0 0 0 0 0  |
     | 0  0 0 0 0 0  -1 0  0  0  0  0 0 0 0 0  |
     | 0  0 0 0 0 0  0  -1 0  0  0  0 0 0 0 0  |
     | 0  0 1 0 0 0  0  0  0  0  0  0 0 0 0 0  |
     | 0  0 0 1 0 0  0  0  0  0  0  0 0 0 0 0  |
     | 0  0 0 0 1 0  0  0  0  0  0  0 0 0 0 0  |
     | 0  0 0 0 0 0  0  0  0  0  0  1 0 0 0 0  |
     | 0  0 0 0 0 0  0  0  0  0  0  0 1 0 0 0  |
     | 0  0 0 0 0 0  0  0  0  0  0  0 0 1 0 0  |
     | 0  0 0 0 0 0  0  0  -1 0  0  0 0 0 0 0  |
     | 0  0 0 0 0 0  0  0  0  -1 0  0 0 0 0 0  |
     | 0  0 0 0 0 0  0  0  0  0  -1 0 0 0 0 0  |
     | 0  0 0 0 0 0  0  0  0  0  0  0 0 0 0 -1 |
     | 0  0 0 0 0 0  0  0  0  0  0  0 0 0 1 0  |

              16       16
o9 : Matrix QQ   <-- QQ

We check that this gives an automorphism of the orthogonal Grassmannian.


i10 : S=QQ[x_0, x_12, x_13, x_14, x_15, x_23, x_24, x_25, x_34, x_35, x_45, x_1234, x_1235, x_1245, x_1345, x_2345];

i11 : OG510 = ideal {x_0*x_2345-x_23*x_45+x_24*x_35-x_25*x_34,
      x_12*x_1345-x_13*x_1245+x_14*x_1235-x_15*x_1234,
      x_0*x_1345-x_13*x_45+x_14*x_35-x_15*x_34,
      x_12*x_2345-x_23*x_1245+x_24*x_1235-x_25*x_1234,
      x_0*x_1245-x_12*x_45+x_14*x_25-x_15*x_24,
      x_13*x_2345-x_23*x_1345+x_34*x_1235-x_35*x_1234,
      x_0*x_1235-x_12*x_35+x_13*x_25-x_15*x_23,
      x_14*x_2345-x_24*x_1345+x_34*x_1245-x_45*x_1234,
      x_0*x_1234-x_12*x_34+x_13*x_24-x_14*x_23,
      x_15*x_2345-x_25*x_1345+x_35*x_1245-x_45*x_1235};

o11 : Ideal of S

i12 : F1 = map(S,S,transpose g1);

o12 : RingMap S <-- S

i13 : F1(OG510)==OG510

o13 = true

Next, we choose a random matrix \(M_{7,1} \in \operatorname{Mat}_{7 \times 16}\) and consider the point in \(\operatorname{Gr}(7,16)\) corresponding to its row space. We compute the value of \(F_{7,1}\) on \(M_{7,1}\) and \(g_1.M_{7,1}\).

i14 : M71 = matrix {{-10, -10, 10, 1, -8, 10, 9, 9, -5, 3, -1, 3, -3, 6, 6, 9}, {9, 9, -3, 1, -4, -5, -4, 9, -5, -1, 0, -7, 0, 2, -9, -9}, {7, 3, -6, 9, -5, -3, 9, 5, 0, -5, 4, -6, 3, 9, -2, 8}, {-1, 2, -9, -4, 9, -5, -1, 7, 10, -1, 10, -3, -8, -7, -7, -5}, {-5, 10, 3, -5, 4, -8, 5, -7, -4, -2, 1, 0, 0, 10, -10, 6}, {-4, -8, 8, 9, 0, 3, 4, -6, 8, -8, 0, -9, 3, 0, 5, -8}, {-8, -3, 6, 2, 6, 9, -5, -3, 5, 2, 1, 9, -1, -3, -4, -9}}

o14 = | -10 -10 10 1  -8 10 9  9  -5 3  -1 3  -3 6  6   9  |
      | 9   9   -3 1  -4 -5 -4 9  -5 -1 0  -7 0  2  -9  -9 |
      | 7   3   -6 9  -5 -3 9  5  0  -5 4  -6 3  9  -2  8  |
      | -1  2   -9 -4 9  -5 -1 7  10 -1 10 -3 -8 -7 -7  -5 |
      | -5  10  3  -5 4  -8 5  -7 -4 -2 1  0  0  10 -10 6  |
      | -4  -8  8  9  0  3  4  -6 8  -8 0  -9 3  0  5   -8 |
      | -8  -3  6  2  6  9  -5 -3 5  2  1  9  -1 -3 -4  -9 |

               7       16
o14 : Matrix ZZ  <-- ZZ

i15 : F71(M71)

o15 = -255458271033377715965762854032954123600

o15 : QQ

i16 : F71(transpose(g1*transpose(M71)))

o16 = -255458271033377715965762854032954123600

o16 : QQ

Next, we check that moving the point in \(\operatorname{Gr}(7,16)\) corresponding to the cuspidal curve of genus 7 with heptagonal symmetry does not change the value of \(F_{7,1}\).

i17 : R=QQ[x_0,x_12,x_13,x_14,x_15,x_23,x_24,x_25,x_34,x_35,x_45,x_1234,x_1235,x_1245,x_1345,x_2345];

i18 : LCusp7 = {x_13, x_23+(5/3)*x_15, x_24, x_34-5*x_25, x_45-(4/3)*x_35, x_1245-(9/8)*x_1235, x_1345+(1/2)*x_12, x_2345+(2/15)*x_14,x_1234-1/30*x_0};

i19 : MCusp7 = matrixFromHyperplaneEquations LCusp7

o19 = | 0  0  0 0     -3/5 1 0 0   0 0   0 0 0   0 0 0 |
      | 0  0  0 0     0    0 0 1/5 1 0   0 0 0   0 0 0 |
      | 0  0  0 0     0    0 0 0   0 3/4 1 0 0   0 0 0 |
      | 30 0  0 0     0    0 0 0   0 0   0 1 0   0 0 0 |
      | 0  0  0 0     0    0 0 0   0 0   0 0 8/9 1 0 0 |
      | 0  -2 0 0     0    0 0 0   0 0   0 0 0   0 1 0 |
      | 0  0  0 -15/2 0    0 0 0   0 0   0 0 0   0 0 1 |

               7       16
o19 : Matrix QQ  <-- QQ

i20 : F71(MCusp7)

      3038765625
o20 = ----------
           2

o20 : QQ

i21 : F71(transpose(g1*transpose(MCusp7)))

      3038765625
o21 = ----------
           2

o21 : QQ

Finally, we randomly chose some additional vectors with norms in \( \mathbb{Q}\). We use these to create two additional elements \(g_2\) and \(g_3\) in \(\operatorname{Spin}(10)\) and check that \(F_{7,1}\) is invariant under these elements as well.

i22 : ReflectionVectorList2 = {1/12*{4, 8, 3, -8, -10, 1, 10, 0, 0, -6}, 1/3*{-1, 5, -5, -1, 10, 3, 6, -5, 3, -4}};

i23 : all(ReflectionVectorList2, v -> Q(v)==1)

o23 = true

i24 : g2 = actionOnSplus(ReflectionVectorList2,FullContractAndWedgeList,Splusindices)

o24 = | 61/36  -7/9   17/36  1/3    -5/6   55/36  -8/9   -65/18 43/36  5/9   5/2    0      0      0      0      0      |
      | -2/3   25/18  -17/9  19/18  40/9   7/18   -23/36 -10/9  0      0     0      43/36  5/9    5/2    0      0      |
      | -5/36  10/9   7/4    -10/9  -25/18 -19/36 0      0      -23/36 -10/9 0      8/9    65/18  0      5/2    0      |
      | 1/12   -2/3   -1/4   2      5/6    0      -19/36 0      -7/18  0     -10/9  55/36  0      65/18  -5/9   0      |
      | 7/18   1/18   7/6    -13/18 -13/9  0      0      -19/36 0      -7/18 23/36  0      55/36  -8/9   43/36  0      |
      | -25/18 -5/9   -17/18 0      0      13/6   -10/9  -25/18 -19/18 -40/9 0      1/3    -5/6   0      0      5/2    |
      | 5/6    1/3    0      -17/18 0      -1/4   29/12  5/6    -17/9  0     -40/9  -17/36 0      -5/6   0      -5/9   |
      | -1/9   -11/18 0      0      -17/18 7/6    -13/18 -37/36 0      -17/9 19/18  0      -17/36 -1/3   0      43/36  |
      | 0      0      1/3    5/9    0      2/3    10/9   0      25/9   5/6   25/18  -7/9   0      0      -5/6   -65/18 |
      | -5/6   0      -11/18 0      5/9    -1/18  0      10/9   -13/18 -2/3  -10/9  0      -7/9   0      -1/3   8/9    |
      | 1/2    0      0      -11/18 -1/3   0      -1/18  -2/3   -7/6   -1/4  -5/12  0      0      -7/9   17/36  55/36  |
      | 0      0      -5/6   -25/18 0      1/12   5/36   0      -2/3   0     0      89/36  5/6    25/18  40/9   -10/9  |
      | 0      -5/6   1/9    0      -25/18 7/18   0      5/36   0      -2/3  0      -13/18 -35/36 -10/9  -19/18 23/36  |
      | 0      1/2    0      1/9    5/6    0      7/18   -1/12  0      0     -2/3   -7/6   -1/4   -13/18 -17/9  7/18   |
      | 0      0      1/2    5/6    0      0      0      0      7/18   -1/12 -5/36  1/18   2/3    10/9   -13/36 -19/36 |
      | 0      0      0      0      0      1/2    5/6    0      -1/9   -5/6  -25/18 -11/18 -1/3   -5/9   -17/18 1/18   |

               16       16
o24 : Matrix QQ   <-- QQ

i25 : F2 = map(S,S,transpose g2);

o25 : RingMap S <-- S

i26 : F2(OG510)==OG510

o26 = true

i27 : F71(transpose(g2*transpose(M71)))

o27 = -255458271033377715965762854032954123600

o27 : QQ

i28 : F71(transpose(g2*transpose(MCusp7)))

      3038765625
o28 = ----------
           2

o28 : QQ

i29 : ReflectionVectorList3 = {1/6*{-1, -4, 6, 10, -1, 1, 8, 10, 0, -9}, 1/6*{-3, 8, 0, 9, 1, -3, -1, -3, 5, -10}};

i30 : all(ReflectionVectorList3, v -> Q(v)==1)

o30 = true

i31 : g3 = actionOnSplus(ReflectionVectorList3,FullContractAndWedgeList,Splusindices)

o31 = | 49/36   5/9     -1/2   -7/12  1/9    4/3    29/9   -1/9  -3/2   -1/6  -19/36 0      0    0      0      0      |
      | 23/36   103/36  1/6    41/18  7/36   -1/2   -13/12 1/18  0      0     0      -3/2   -1/6 -19/36 0      0      |
      | 3/4     17/9    61/36  10/3   7/36   -1/9   0      0     -13/12 1/18  0      -29/9  1/9  0      -19/36 0      |
      | 5/36    5/9     -5/6   -7/36  5/36   0      -1/9   0     1/2    0     1/18   4/3    0    1/9    1/6    0      |
      | -37/36  -28/9   5/3    19/36  2/3    0      0      -1/9  0      1/2   13/12  0      4/3  29/9   -3/2   0      |
      | -7/18   11/12   -25/36 0      0      127/36 10/3   7/36  -41/18 -7/36 0      -7/12  1/9  0      0      -19/36 |
      | 10/9    -5/36   0      -25/36 0      -5/6   59/36  5/36  1/6    0     -7/36  1/2    0    1/9    0      1/6    |
      | -89/36  -17/36  0      0      -25/36 5/3    19/36  5/2   0      1/6   41/18  0      1/2  7/12   0      -3/2   |
      | 25/18   0       -5/36  -11/12 0      -5/9   17/9   0     17/36  5/36  -7/36  5/9    0    0      1/9    -1/9   |
      | -127/36 0       -17/36 0      -11/12 28/9   0      17/9  19/36  4/3   10/3   0      5/9  0      7/12   -29/9  |
      | 5/4     0       0      -17/36 5/36   0      28/9   5/9   -5/3   -5/6  -5/9   0      0    5/9    -1/2   4/3    |
      | 0       25/18   -10/9  -7/18  0      5/36   -3/4   0     23/36  0     0      71/36  5/36 -7/36  7/36   1/18   |
      | 0       -127/36 89/36  0      -7/18  -37/36 0      -3/4  0      23/36 0      19/36  17/6 10/3   -41/18 13/12  |
      | 0       5/4     0      89/36  10/9   0      -37/36 -5/36 0      0     23/36  -5/3   -5/6 17/18  1/6    -1/2   |
      | 0       0       5/4    127/36 25/18  0      0      0     -37/36 -5/36 3/4    -28/9  -5/9 17/9   -2/9   -1/9   |
      | 0       0       0      0      0      5/4    127/36 25/18 -89/36 -10/9 -7/18  -17/36 5/36 11/12  -25/36 29/18  |

               16       16
o31 : Matrix QQ   <-- QQ

i32 : F3 = map(S,S,transpose g3);

o32 : RingMap S <-- S

i33 : F3(OG510)==OG510

o33 = true

i34 : F71(transpose(g3*transpose(M71)))

o34 = -255458271033377715965762854032954123600

o34 : QQ

i35 : F71(transpose(g3*transpose(MCusp7)))

      3038765625
o35 = ----------
           2

o35 : QQ

Testing the invariance of \(F_{7,2}\)

We test the invariance of \(F_{7,2}\). We continue to use the elements \(g_1,g_2,g_3 \in \operatorname{Spin}(10) \) constructed above. We choose a random matrix \(M_{7,2} \in \operatorname{Mat}_{8 \times 16}\) and consider the point in \(\operatorname{Gr}(8,16)\) corresponding to its row space. We also study the point in \(\operatorname{Gr}(8,16)\) corresponding to the tangent developable of genus 7.

i36 : M72 = matrix {{-10, -10, 10, 1, -8, 10, 9, 9, -5, 3, -1, 3, -3, 6, 6, 9}, {9, 9, -3, 1, -4, -5, -4, 9, -5, -1, 0, -7, 0, 2, -9, -9}, {7, 3, -6, 9, -5, -3, 9, 5, 0, -5, 4, -6, 3, 9, -2, 8}, {-1, 2, -9, -4, 9, -5, -1, 7, 10, -1, 10, -3, -8, -7, -7, -5}, {-5, 10, 3, -5, 4, -8, 5, -7, -4, -2, 1, 0, 0, 10, -10, 6}, {-4, -8, 8, 9, 0, 3, 4, -6, 8, -8, 0, -9, 3, 0, 5, -8}, {-8, -3, 6, 2, 6, 9, -5, -3, 5, 2, 1, 9, -1, -3, -4, -9}, {-2, 9, -9, -6, 7, 2, 0, 0, 2, 3, -7, -6, -1, -2, 3, 3}}

o36 = | -10 -10 10 1  -8 10 9  9  -5 3  -1 3  -3 6  6   9  |
      | 9   9   -3 1  -4 -5 -4 9  -5 -1 0  -7 0  2  -9  -9 |
      | 7   3   -6 9  -5 -3 9  5  0  -5 4  -6 3  9  -2  8  |
      | -1  2   -9 -4 9  -5 -1 7  10 -1 10 -3 -8 -7 -7  -5 |
      | -5  10  3  -5 4  -8 5  -7 -4 -2 1  0  0  10 -10 6  |
      | -4  -8  8  9  0  3  4  -6 8  -8 0  -9 3  0  5   -8 |
      | -8  -3  6  2  6  9  -5 -3 5  2  1  9  -1 -3 -4  -9 |
      | -2  9   -9 -6 7  2  0  0  2  3  -7 -6 -1 -2 3   3  |

               8       16
o36 : Matrix ZZ  <-- ZZ

i37 : LTD7 = {x_13, x_23+(5/3)*x_15, x_24, x_34-5*x_25, x_45-(4/3)*x_35, x_1245-(9/8)*x_1235, x_1345+(1/2)*x_12, x_2345+(2/15)*x_14};

i38 : MTD7 = matrixFromHyperplaneEquations LTD7

o38 = | 1 0  0 0     0    0 0 0   0 0   0 0 0   0 0 0 |
      | 0 0  0 0     -3/5 1 0 0   0 0   0 0 0   0 0 0 |
      | 0 0  0 0     0    0 0 1/5 1 0   0 0 0   0 0 0 |
      | 0 0  0 0     0    0 0 0   0 3/4 1 0 0   0 0 0 |
      | 0 0  0 0     0    0 0 0   0 0   0 1 0   0 0 0 |
      | 0 0  0 0     0    0 0 0   0 0   0 0 8/9 1 0 0 |
      | 0 -2 0 0     0    0 0 0   0 0   0 0 0   0 1 0 |
      | 0 0  0 -15/2 0    0 0 0   0 0   0 0 0   0 0 1 |

               8       16
o38 : Matrix QQ  <-- QQ

i39 : F72(M72)

o39 = 17163114907498626573888

o39 : QQ

i40 : F72(transpose(g1*transpose(M72)))

o40 = 17163114907498626573888

o40 : QQ

i41 : F72(transpose(g2*transpose(M72)))

o41 = 17163114907498626573888

o41 : QQ

i42 : F72(transpose(g3*transpose(M72)))

o42 = 17163114907498626573888

o42 : QQ

i43 : F72(MTD7)

      665
o43 = ---
       2

o43 : QQ

i44 : F72(transpose(g1*transpose(MTD7)))

      665
o44 = ---
       2

o44 : QQ

i45 : F72(transpose(g2*transpose(MTD7)))

      665
o45 = ---
       2

o45 : QQ

i46 : F72(transpose(g3*transpose(MTD7)))

      665
o46 = ---
       2

o46 : QQ