Fordham University

Testing the invariance of \(F_{10,1}\) and \(F_{10,2}\)

The polynomials \(F_{10,1}\) and \(F_{10,2}\) are constructed to be invariant under the action of the algebraic group \(G_2\) on its adjoint representation. We check this below as follows.

We generate elements of \(G_2\) as follows. First, we construct the basis \(\mathscr{B}\) of the Lie algebra \(\mathfrak{g}_2\) described by Fulton and Harris in Representation Theory, Section 22.1. Then we compute the matrices \(\rho(B_i)\) giving the adjoint representation of \(\mathfrak{g}_2\). Finally, we compute \(\operatorname{exp}(c\rho(B_i)) \in G_2\) for each element \(B_i \in \mathscr{B}.\)

We were partly inspired by Ruben Arenas's 2005 senior thesis Constructing a Matrix Representation of the Lie Group \(\mathit{G}_2\) at Harvey Mudd College, which applies this strategy of exponentiating elements of a basis of \(\mathfrak{g}_2\) to obtain elements of \(G_2\). The results in this thesis illustrate that the elements produced by this method are elegant.

The elements \(B_0\) and \(B_1\) span the Cartan subalgebra \( \mathfrak{h} \subset \mathfrak{g}_2\), and \(\rho(B_0)\) and \(\rho(B_1)\) are diagonal matrices. Let \(a_0,\ldots,a_{13}\) be the diagonal entries of \(B_i\) for \(i=0\) or 1. Then \(\exp(c B_i)\) is the diagonal matrix with entries \(e^{ca_0},\ldots,e^{ca_{13}}\). To keep our computations in \(\mathbb{Q}\) we make a logarithmic change of variables \(c = \ln(t)\) and work with the matrix with diagonal entries \(t^{a_i}\) instead.

For \(i=2,\ldots,13\), we check below that the matrices \(\rho(B_i)\) satisfy \( \rho(B_i)^4=0.\) Thus, we can compute \(\operatorname{exp}(c\rho(B_i))\) using the first four terms in the Taylor series for the exponential map.

Testing the invariance of \(F_{10,1}\)

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
     );

i4 : needsPackage("LieAlgebraRepresentations");
Next, we construct the Lie algebra \(\mathfrak{g}_2\) and its adjoint representation. 
i5 : g2=simpleLieAlgebra("G",2);

i6 : V01=adjointRepresentation(g2);

i7 : L = V01#"RepresentationMatrices";
We view the matrix \(\rho(X_1)\) and check that the basis elements outside the Cartan subalgebra satisfy \(M^4 =0\). 
i8 : L_2

o8 = | 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 |
     | -2 1 0 0 0 0  0 0 0 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 2 0  0 0 0 0 0  0  0 0 |
     | 0  0 0 0 0 -3 0 0 0 0 0  0  0 0 |
     | 0  0 0 0 0 0  0 0 0 0 0  0  0 0 |
     | 0  0 0 0 0 0  0 0 0 0 0  0  0 0 |
     | 0  0 0 0 0 0  0 0 0 0 -3 0  0 0 |
     | 0  0 0 0 0 0  0 0 0 0 0  -2 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 0 0 0 0  0  0 0 |

              14       14
o8 : Matrix QQ   <-- QQ

i9 : L_2^4

o9 = 0

              14       14
o9 : Matrix QQ   <-- QQ

i10 : all(toList(2..13), i -> L_i^4==0)

o10 = true
We compute the diagonal matrices \(\operatorname{exp}(\rho(B_i))\) discussed above and view \(\operatorname{exp}(\rho(H_1))\). 
i11 : R=frac(QQ[c,t,Degrees=>{0,0}]);

i12 : G2elementsa = apply(2, i -> diagonalMatrix(apply(14, j ->t^(lift((L_i)_(j,j),ZZ)))));

i13 : first G2elementsa

o13 = | 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 t2 0    0   0 0  0 0    0  0 0   0    0 |
      | 0 0 0  1/t3 0   0 0  0 0    0  0 0   0    0 |
      | 0 0 0  0    1/t 0 0  0 0    0  0 0   0    0 |
      | 0 0 0  0    0   t 0  0 0    0  0 0   0    0 |
      | 0 0 0  0    0   0 t3 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/t2 0  0 0   0    0 |
      | 0 0 0  0    0   0 0  0 0    t3 0 0   0    0 |
      | 0 0 0  0    0   0 0  0 0    0  t 0   0    0 |
      | 0 0 0  0    0   0 0  0 0    0  0 1/t 0    0 |
      | 0 0 0  0    0   0 0  0 0    0  0 0   1/t3 0 |
      | 0 0 0  0    0   0 0  0 0    0  0 0   0    1 |

              14      14
o13 : Matrix R   <-- R
Next we introduce a function to compute the sum of the first \(n\) terms in the Taylor series of the exponential function, and use this to compute \(\operatorname{exp}(c\rho(B_i))\) for the basis elements outside the Cartan subalgebra. We also view \(\operatorname{exp}(c\rho(X_1))\). 
i14 : truncatedMatrixExponential = (n,M) -> sum apply(n, i -> M^i/(i!));

i15 : G2elementsb = apply(toList(2..13), i -> truncatedMatrixExponential(4,c*(L_i)));

i16 : first G2elementsb

o16 = | 1   0 0 0   0    0   0 0 c   0 0   0   0   0 |
      | 0   1 0 0   0    0   0 0 0   0 0   0   0   0 |
      | -2c c 1 0   0    0   0 0 -c2 0 0   0   0   0 |
      | 0   0 0 1   0    0   0 0 0   0 0   0   0   0 |
      | 0   0 0 c   1    0   0 0 0   0 0   0   0   0 |
      | 0   0 0 c2  2c   1   0 0 0   0 0   0   0   0 |
      | 0   0 0 -c3 -3c2 -3c 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 1   0 0   0   0   0 |
      | 0   0 0 0   0    0   0 0 0   1 -3c 3c2 c3  0 |
      | 0   0 0 0   0    0   0 0 0   0 1   -2c -c2 0 |
      | 0   0 0 0   0    0   0 0 0   0 0   1   c   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 |

              14      14
o16 : Matrix R   <-- R

i17 : G2elements = join(G2elementsa,G2elementsb);
We check that the adjoint variety for \(G_2\) is invariant under the action of these elements. 
i18 : S = R[H_1,H_2,X_1,X_2,X_3,X_4,X_5,X_6,Y_1,Y_2,Y_3,Y_4,Y_5,Y_6];

i19 : G2AdjVar = ideal {Y_4^2+Y_3*Y_5-Y_1*Y_6, Y_3*Y_4+Y_2*Y_5+2*H_1*Y_6-H_2*Y_6, Y_1*Y_4-H_1*Y_5+H_2*Y_5+X_2*Y_6, H_1*Y_4+X_1*Y_5+X_3*Y_6, Y_3^2-Y_2*Y_4+X_1*Y_6, Y_1*Y_3-H_2*Y_4-X_1*Y_5-2*X_3*Y_6, H_1*Y_3-X_1*Y_4-X_4*Y_6, Y_1*Y_2-H_2*Y_3+X_1*Y_4+2*X_4*Y_6, H_1*Y_2-X_1*Y_3-X_5*Y_6, Y_1^2+X_2*Y_4-X_3*Y_5, X_1*Y_1+X_3*Y_3+X_4*Y_4, H_2*Y_1+X_2*Y_3+2*X_3*Y_4-X_4*Y_5, H_1*Y_1+X_3*Y_4-X_4*Y_5, X_4^2+X_3*X_5-X_1*X_6, X_3*X_4+X_2*X_5+2*H_1*X_6-H_2*X_6, X_1*X_4-H_1*X_5+H_2*X_5+X_6*Y_2, H_1*X_4+X_5*Y_1+X_6*Y_3, X_3^2-X_2*X_4+X_6*Y_1, X_1*X_3-H_2*X_4-X_5*Y_1-2*X_6*Y_3, H_1*X_3-X_4*Y_1-X_6*Y_4, X_1*X_2-H_2*X_3+X_4*Y_1+2*X_6*Y_4, H_1*X_2-X_3*Y_1-X_6*Y_5, X_1^2+X_4*Y_2-X_5*Y_3, H_2*X_1+X_3*Y_2+2*X_4*Y_3-X_5*Y_4, H_1*X_1+X_4*Y_3-X_5*Y_4, H_2^2+X_2*Y_2+3*X_3*Y_3+3*X_4*Y_4+X_5*Y_5+4*X_6*Y_6, H_1*H_2+X_3*Y_3+2*X_4*Y_4+X_5*Y_5+2*X_6*Y_6, H_1^2+X_4*Y_4+X_5*Y_5+X_6*Y_6};

o19 : Ideal of S

i20 : G2elementmaps = apply(G2elements, g -> map(S,S,transpose g));

i21 : all(G2elementmaps, F -> F(G2AdjVar)==G2AdjVar)

o21 = true

Next, we choose a random matrix \(M_{10,1} \in \operatorname{Mat}_{10 \times 14}\) and consider the point in \(\operatorname{Gr}(10,14)\) corresponding to its row space. We compute the value of \(F_{10,1}\) on \(M_{10,1}\) and check that it takes the same value on \(g.M_{10,1}\) for each \(g \in G_2\) that we constructed. 

i22 : M101=matrix {{-2, -2, 5, 9, 6, -4, 5, -2, 8, 3, -8, -8, 4, -1}, {-7, 1, 2, 3, -7, 8, 9, -9, 3, -4, -8, 4, 10, 10}, {10, 3, 5, 1, 10, -5, -3, 9, 1, 8, -5, 0, 6, 5}, {-4, -1, 4, 9, -8, 8, 5, 0, 4, 3, 1, -6, 0, 0}, {1, 2, 0, 10, 6, 8, 10, 8, 9, 4, 8, -3, -8, 2}, {-1, -3, -7, -7, 2, 7, -5, 3, 0, 2, -3, -3, 9, -2}, {4, -9, 5, -7, 10, -3, -10, -4, -8, -9, -5, 7, 8, -7}, {-7, -10, 3, 2, 10, -4, -10, 10, 8, 9, -9, 3, 1, -6}, {-7, 1, -8, 9, -2, -2, 2, 8, -3, 7, -7, 0, -10, -1}, {-4, -9, -2, -3, -8, -7, 0, 5, 0, 4, 5, -8, -8, -7}}

o22 = | -2 -2  5  9  6  -4 5   -2 8  3  -8 -8 4   -1 |
      | -7 1   2  3  -7 8  9   -9 3  -4 -8 4  10  10 |
      | 10 3   5  1  10 -5 -3  9  1  8  -5 0  6   5  |
      | -4 -1  4  9  -8 8  5   0  4  3  1  -6 0   0  |
      | 1  2   0  10 6  8  10  8  9  4  8  -3 -8  2  |
      | -1 -3  -7 -7 2  7  -5  3  0  2  -3 -3 9   -2 |
      | 4  -9  5  -7 10 -3 -10 -4 -8 -9 -5 7  8   -7 |
      | -7 -10 3  2  10 -4 -10 10 8  9  -9 3  1   -6 |
      | -7 1   -8 9  -2 -2 2   8  -3 7  -7 0  -10 -1 |
      | -4 -9  -2 -3 -8 -7 0   5  0  4  5  -8 -8  -7 |

               10       14
o22 : Matrix ZZ   <-- ZZ

i23 : a1=F101(M101)

      587970911896663267527128
o23 = ------------------------
                  3

o23 : QQ

i24 : all(G2elements, g -> F101(transpose(g*transpose(M101)))==a1)

o24 = true
Next, we check that moving the point in \(\operatorname{Gr}(10,14)\) corresponding to the cuspidal curve of genus 10 with decagonal symmetry does not change the value of \(F_{10,1}\). 
i25 : LCusp10={H_1-4/9*H_2, X_3-4/27*Y_6, Y_3-5/4*X_6,X_5+8/75*Y_5};

i26 : MCusp10 = matrixFromHyperplaneEquations LCusp10

o26 = | 4/9 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 1 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 1 0 0 0 0 |
      | 0   0 0 0 0    0 0     4/5 0 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 -8/75 0   0 0 0 0 1 0 |
      | 0   0 0 0 4/27 0 0     0   0 0 0 0 0 1 |

              10      14
o26 : Matrix R   <-- R

i27 : a2=F101(MCusp10)

       25088
o27 = -------
      1366875

o27 : R

i28 : all(G2elements, g -> F101(transpose(g*transpose(MCusp10)))==a2)

o28 = true

Testing the invariance of \(F_{10,2}\)

We test the invariance of \(F_{10,2}\). We continue to use the elements \(g \in G_2 \) constructed above. We choose a random matrix \(M_{10,2} \in \operatorname{Mat}_{11 \times 14}\) and consider the point in \(\operatorname{Gr}(11,14)\) corresponding to its row space. We also study the point in \(\operatorname{Gr}(11,14)\) corresponding to the tangent developable of genus 10. 
i29 : M102=matrix {{-2, -2, 5, 9, 6, -4, 5, -2, 8, 3, -8, -8, 4, -1}, {-7, 1, 2, 3, -7, 8, 9, -9, 3, -4, -8, 4, 10, 10}, {10, 3, 5, 1, 10, -5, -3, 9, 1, 8, -5, 0, 6, 5}, {-4, -1, 4, 9, -8, 8, 5, 0, 4, 3, 1, -6, 0, 0}, {1, 2, 0, 10, 6, 8, 10, 8, 9, 4, 8, -3, -8, 2}, {-1, -3, -7, -7, 2, 7, -5, 3, 0, 2, -3, -3, 9, -2}, {4, -9, 5, -7, 10, -3, -10, -4, -8, -9, -5, 7, 8, -7}, {-7, -10, 3, 2, 10, -4, -10, 10, 8, 9, -9, 3, 1, -6}, {-7, 1, -8, 9, -2, -2, 2, 8, -3, 7, -7, 0, -10, -1}, {-4, -9, -2, -3, -8, -7, 0, 5, 0, 4, 5, -8, -8, -7}, {-4, 10, 8, -3, -1, 8, -2, 1, 4, 1, 4, -6, 4, 7}}

o29 = | -2 -2  5  9  6  -4 5   -2 8  3  -8 -8 4   -1 |
      | -7 1   2  3  -7 8  9   -9 3  -4 -8 4  10  10 |
      | 10 3   5  1  10 -5 -3  9  1  8  -5 0  6   5  |
      | -4 -1  4  9  -8 8  5   0  4  3  1  -6 0   0  |
      | 1  2   0  10 6  8  10  8  9  4  8  -3 -8  2  |
      | -1 -3  -7 -7 2  7  -5  3  0  2  -3 -3 9   -2 |
      | 4  -9  5  -7 10 -3 -10 -4 -8 -9 -5 7  8   -7 |
      | -7 -10 3  2  10 -4 -10 10 8  9  -9 3  1   -6 |
      | -7 1   -8 9  -2 -2 2   8  -3 7  -7 0  -10 -1 |
      | -4 -9  -2 -3 -8 -7 0   5  0  4  5  -8 -8  -7 |
      | -4 10  8  -3 -1 8  -2  1  4  1  4  -6 4   7  |

               11       14
o29 : Matrix ZZ   <-- ZZ

i30 : a3=F102(M102)

o30 = -28637042462048

i31 : all(G2elements, g -> F102(transpose(g*transpose(M102)))==a3)

o31 = true

i32 : S = QQ[H_1,H_2,X_1,X_2,X_3,X_4,X_5,X_6,Y_1,Y_2,Y_3,Y_4,Y_5,Y_6];

i33 : LTD10={H_1-4/9*H_2, X_3-4/27*Y_6, Y_3-5/4*X_6};

i34 : MTD10 = matrixFromHyperplaneEquations LTD10;

               11       14
o34 : Matrix QQ   <-- QQ

i35 : a4=F102(MTD10)

         56
o35 = - ---
        135

o35 : QQ

i36 : all(G2elements, g -> F102(transpose(g*transpose(MTD10)))==a4)

o36 = true