Testing the invariance of \(F_{9,1}\) and \(F_{9,2}\)

The polynomials \(F_{9,1}\) and \(F_{9,2}\) are constructed to be invariant under the action of \(\operatorname{Sp}(6)\) on the 14-dimensional irreducible representation \(V(0,0,1)\) whose highest weight is the fundamental dominant weight \(\omega_3\). We check this below as follows.

The symplectic group is generated by transformations known as symplectic transvections. See below for a definition and a reference.

We test that \(F_{9,1}\) and \(F_{9,2}\) are invariant under some transvections and their products. First, we test this for a transvection whose inputs are simple enough that the action can be checked by hand. Then, we test invariance under transvections with respect to random vectors.

Testing the invariance of \(F_{9,1}\)

Let \(B\) be a symplectic form on a vector space \(V\) of dimension \(2n\). That is, \(B\) is a nondegenerate alternating form.

Let \(c\) be a scalar, and let \(w\) be a vector in \(V\). A symplectic transvection is a transformation of the form \( \tau(v) = v+c B(v,w) w\) for some scalar \(c\) and some vector \(w \in V\). These generate the symplectic group. See for instance Artin, Geometric Algebra, Section III.5, especially formula (3.41), Definition 3.14, and Theorem 3.25.

We use the alternating form \(B\) corresponding to the matrix \[ \left[ \begin{array}{rr} 0 & I_n \\ -I_n & 0 \end{array}\right]. \]

First, consider the symplectic transformation \(g_1\) corresponding to the choices \(c = 1\) and \(w = e_1\). We have \(B(e_i,w)=0\) when \(i \neq 4\), and \(B(e_4,w) = -1\). Thus the matrix of \(g_1\) is \[ \left[ \begin{array}{rrrrrr} 1 & 0 & 0 & -1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array} \right]. \] The basis we use for the 14-dimensional vector space \(V(0,0,1)\) is as follows. We write \(e_{i_{1}i_{2}i_{3}}\) for the wedge product \(e_{i_1} \wedge e_{i_2} \wedge e_{i_3}\). \[ \mathscr{B} = \{ e_{123}, e_{234},e_{134}-e_{235},-e_{124}-e_{236}, -e_{135},e_{125}-e_{136}, e_{126}, e_{156}, e_{146}-e_{256}, -e_{145}-e_{356}, -e_{246}, e_{245}-e_{346},e_{345},e_{456}\}. \] The action of \(g_1\) on this basis is as follows. \[ \begin{array}{rcl} g_1.e_{123} &= & e_{123}\\ g_1.e_{234} &= & -e_{123}+ e_{234}\\ g_1.(e_{134}-e_{235}) &= & e_{134}-e_{235}\\ g_1.(-e_{124}-e_{236}) &= & -e_{124}-e_{236}\\ g_1.(-e_{135}) &= & -e_{135}\\ g_1.(e_{125}-e_{136}) &= & e_{125}-e_{136}\\ g_1.e_{126} &= & e_{126}\\ g_1.e_{156} &= & e_{156}\\ g_1.(e_{146}-e_{256}) &= & e_{146}-e_{256}\\ g_1.(-e_{145}-e_{356}) &= & -e_{145}-e_{356}\\ g_1.(e_{245}-e_{346}) &= & e_{125}-e_{136}+e_{245}-e_{346}\\ g_1.e_{345} &= & e_{135}+e_{345}\\ g_1.e_{456} &= & -e_{156}+e_{456} \end{array} \] This matches the matrix rhog1 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 "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 : Q = FHsymplecticFormMatrix(3,QQ);

              6       6
o4 : Matrix QQ  <-- QQ

We enter \(c=1 \) and \(w=e_1\), and compute the element \(g_1 \in \operatorname{Sp}(6)\).

i5 : c1 = 1

o5 = 1

i6 : v1 = {1,0,0,0,0,0}

o6 = {1, 0, 0, 0, 0, 0}

o6 : List

i7 : g1 = symplecticTransvectionMatrix(Q,c1,v1)

o7 = | 1 0 0 -1 0 0 |
     | 0 1 0 0  0 0 |
     | 0 0 1 0  0 0 |
     | 0 0 0 1  0 0 |
     | 0 0 0 0  1 0 |
     | 0 0 0 0  0 1 |

              6       6
o7 : Matrix QQ  <-- QQ

i8 : isInFultonHarrisSp2n(g1)

o8 = true

i9 : rhog1 = restrictToSubspace(exteriorPower(3,g1),MyBasisVw3C3)

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

              14       14
o9 : Matrix QQ   <-- QQ

We check that the symplectic Grassmannian is invariant under \(g_1\).

i10 : S = QQ[z_0..z_13];

i11 : I1 = {-z_5^2+z_4*z_6-z_0*z_7,
       z_3*z_5-z_2*z_6-z_0*z_8,
       -z_3*z_4+z_2*z_5-z_0*z_9,
       -z_3^2+z_1*z_6-z_0*z_10,
       z_2*z_3-z_1*z_5-z_0*z_11,
       -z_2^2+z_1*z_4-z_0*z_12};

i12 : I2 = {-z_11^2+z_10*z_12-z_1*z_13,
       z_9*z_11-z_8*z_12-z_2*z_13,
       -z_9*z_10+z_8*z_11-z_3*z_13,
       -z_9^2+z_7*z_12-z_4*z_13,
       z_8*z_9-z_7*z_11-z_5*z_13,
       -z_8^2+z_7*z_10-z_6*z_13};

i13 : I3 = {z_1*z_7+z_2*z_8+z_3*z_9-z_0*z_13,
       z_1*z_8+z_2*z_10+z_3*z_11,
       z_1*z_9+z_2*z_11+z_3*z_12,
       z_2*z_7+z_4*z_8+z_5*z_9,
       z_2*z_8+z_4*z_10+z_5*z_11-z_0*z_13,
       z_2*z_9+z_4*z_11+z_5*z_12,
       z_3*z_7+z_5*z_8+z_6*z_9,
       z_3*z_8+z_5*z_10+z_6*z_11,
       z_3*z_9+z_5*z_11+z_6*z_12-z_0*z_13};

i14 : SpGr36 = ideal(flatten {I1,I2,I3});

o14 : Ideal of S

i15 : F1 = map(S,S,transpose rhog1);

o15 : RingMap S <-- S

i16 : F1(SpGr36)==SpGr36

o16 = true

Next, we choose a random matrix \(M_{9,1} \in \operatorname{Mat}_{9 \times 14}\) and consider the point in \(\operatorname{Gr}(9,14)\) corresponding to its row space. We compute the value of \(F_{9,1}\) on \(M_{9,1}\) and \(g_1.M_{9,1}\).

i17 : 
      M91 = matrix {{8, 10, 2, 0, 7, 6, 9, 5, -9, -10, -7, 6, 7, 3}, {1, 0, -8, 4, -4, -4, 6, -5, -2, 7, -4, 7, 9, 2}, {-3, 8, 1, -6, 3, -2, -5, -7, 9, 5, -9, -6, 5, -9}, {-8, -6, 9, 2, -10, 2, 4, 4, 0, 3, -8, -7, -10, -1}, {-1, 8, -3, -1, -9, 0, 10, 10, 8, -10, 4, 6, 4, -10}, {-8, 3, -6, 9, -4, 6, 6, 9, 5, 10, 2, 0, -7, 0}, {7, -3, 9, -2, 1, 4, -6, 7, 6, -6, 0, -10, -6, 8}, {-5, 3, 7, 9, -10, 2, 8, -7, 9, 3, -1, 2, -2, 10}, {-2, -4, 9, -7, 5, 10, -6, 1, 8, 4, 0, 10, 6, -7}}

o17 = | 8  10 2  0  7   6  9  5  -9 -10 -7 6   7   3   |
      | 1  0  -8 4  -4  -4 6  -5 -2 7   -4 7   9   2   |
      | -3 8  1  -6 3   -2 -5 -7 9  5   -9 -6  5   -9  |
      | -8 -6 9  2  -10 2  4  4  0  3   -8 -7  -10 -1  |
      | -1 8  -3 -1 -9  0  10 10 8  -10 4  6   4   -10 |
      | -8 3  -6 9  -4  6  6  9  5  10  2  0   -7  0   |
      | 7  -3 9  -2 1   4  -6 7  6  -6  0  -10 -6  8   |
      | -5 3  7  9  -10 2  8  -7 9  3   -1 2   -2  10  |
      | -2 -4 9  -7 5   10 -6 1  8  4   0  10  6   -7  |

               9       14
o17 : Matrix ZZ  <-- ZZ

i18 : F91(M91)

o18 = -69066628702801276747468144869007195496549388986880

o18 : QQ

i19 : F91(transpose(rhog1*transpose(M91)))

o19 = -69066628702801276747468144869007195496549388986880

o19 : QQ

Next, we construct some additional elements in \( \operatorname{Sp}(6)\) by choosing random scalars and vectors and using them to construct transvections.

i20 : c2 = 1/10;

i21 : v2 = {1, 8/9, 9/5, 9, 7/10, 7/10};

i22 : g2 = symplecticTransvectionMatrix(Q,c2,v2)

o22 = | 19/10  7/100   7/100   -1/10  -4/45   -9/50   |
      | 4/5    239/225 14/225  -4/45  -32/405 -4/25   |
      | 81/50  63/500  563/500 -9/50  -4/25   -81/250 |
      | 81/10  63/100  63/100  1/10   -4/5    -81/50  |
      | 63/100 49/1000 49/1000 -7/100 211/225 -63/500 |
      | 63/100 49/1000 49/1000 -7/100 -14/225 437/500 |

               6       6
o22 : Matrix QQ  <-- QQ

i23 : isInFultonHarrisSp2n(g2)

o23 = true

i24 : rhog2 = restrictToSubspace(exteriorPower(3,g2),MyBasisVw3C3)

o24 = | 9397/4500 -1/10     8/45     9/25     -32/405   -8/25   -81/250   0         0       0       0        0      0        0         |
      | 81/10     1297/4500 8/5      81/25    0         0       0         0         0       0       -81/250  8/25   -32/405  0         |
      | -63/100   7/100     563/500  -63/500  4/5       81/50   0         0         81/250  -4/25   0        9/50   -4/45    0         |
      | -63/100   7/100     -14/225  239/225  0         4/5     81/50     0         -4/25   32/405  -9/50    4/45   0        0         |
      | 49/1000   0         7/50     0        8837/4500 -63/250 0         -81/250   0       -9/25   0        0      -1/10    0         |
      | 49/1000   0         7/100    7/100    -14/225   19/10   -63/500   4/25      9/50    4/45    0        1/10   0        0         |
      | 49/1000   0         0        7/50     0         -28/225 8263/4500 -32/405   -8/45   0       -1/10    0      0        0         |
      | 0         0         0        0        49/1000   -49/500 49/1000   7703/4500 -7/50   -7/50   0        0      0        -1/10     |
      | 0         0         -49/1000 49/1000  0         -63/100 63/100    -4/5      437/500 14/225  -7/100   -7/100 0        4/45      |
      | 0         0         49/1000  -49/1000 63/100    -63/100 0         -81/50    63/500  211/225 0        -7/100 -7/100   9/50      |
      | 0         49/1000   0        63/50    0         0       81/10     0         -8/5    0       163/4500 28/225 0        -32/405   |
      | 0         -49/1000  -63/100  -63/100  0         -81/10  0         0         -81/50  -4/5    63/500   1/10   14/225   -4/25     |
      | 0         49/1000   63/50    0        81/10     0       0         0         0       -81/25  0        63/250 737/4500 -81/250   |
      | 0         0         0        0        0         0       0         81/10     -63/50  -63/50  49/1000  49/500 49/1000  -397/4500 |

               14       14
o24 : Matrix QQ   <-- QQ

i25 : c3 = 4;

i26 : v3 = {9/2, 4/3, 5/7, 4, 3/10, 5/7};

i27 : c4 = 8/7

      8
o27 = -
      7

o27 : QQ

i28 : v4 = {4/7, 1/4, 7/3, 3/7, 8/3, 5/8};

i29 : g3 = symplecticTransvectionMatrix(Q,c4,v4)*symplecticTransvectionMatrix(Q,c3,v3)

o29 = | 1438019/15435 222083/25725 363386/21609 -177923/1715 -1425824/46305 -387494/21609 |
      | 66494/2205    14753/3675   68429/12348  -8318/245    -133393/13230  -18614/3087   |
      | 88616/945     22142/1575   25412/1323   -1586/15     -89426/2835    -30116/1323   |
      | 407096/5145   61952/8575   103709/7203  -151021/1715 -407906/15435  -109736/7203  |
      | 130856/1323   34061/2205   179050/9261  -16397/147   -128183/3969   -227266/9261  |
      | 14767/441     3229/735     157345/24696 -1849/49     -59473/5292    -20348/3087   |

               6       6
o29 : Matrix QQ  <-- QQ

i30 : isInFultonHarrisSp2n(g3)

o30 = true

i31 : rhog3 = restrictToSubspace(exteriorPower(3,g3),MyBasisVw3C3)

o31 = | 1588801199/7563150 -3937896/16807      14429651/108045 -2619212/36015  -86311/4410        -603586/64827     -319977856/756315 1092242/27783        5846968/21609    -90158/9261      7824968/16807     -241316/7203      3721/6174           0                   |
      | 3198472/16807      -1605299401/7563150 4341004/36015   -7079272/108045 -288/343           -84928/2401       -56349728/151263  0                    15690448/64827   -11824/1029      309975584/756315  -3902630/64827    -560257/30870       1092242/27783       |
      | 43675574/540225    -5396448/60025      40448/1323      -781358/77175   633238/12005       -1279952/6615     -5371696/108045   -7845224/64827       30116/1323       -5712074/138915  2001736/36015     -474982/2205      -19579718/324135    2923484/21609       |
      | -102143/7203       453541/28812        -42029/12348    257/8575        -440/343           539258/15435      34296/12005       5912/1029            10946/3087       133393/13230     -124444/36015     586559/15435      3355/3087           -45079/9261         |
      | 728552/1225        -5484672/8575       564784/2401     1117248/8575    1087203128/6806835 -336707008/324135 -512072/77175     -1033224632/2268945  -4003472/36015   -25976192/108045 0                 -13102272/12005   -16389624/84035     7824968/16807       |
      | 47816737/432180    -312156/2401        4033439/43218   -3172717/25725  -5082757/259308    6112112/46305     -266602996/540225 65806253/972405      36635546/108045  1238249/46305    6551136/12005     177923/1715       114985/28812        -120658/7203        |
      | 10223215/403368    -3553225/134456     103675/7203     30036/2401      -6050/3087         -478063/43218     13193354/1260525  -186731/92610        -312392/36015    -6710/3087       -209366/16807     -114985/14406     0                   3721/6174           |
      | 332929/24696       0                   -1087645/28812  318504/1715     43109765/1210104   -38259241/129654  44438966/77175    -1267944329/13613670 -3026420/7203    -3354557/43218   -5484672/8575     -624312/2401      -3553225/134456     2253026/84035       |
      | -124055/7203       1087645/57624       -157345/24696   -6247/5145      -47300/7203        16021/441         378592/180075     397958/108045        4912/1029        495511/37044     -461506/180075    34201/882         103675/14406        -1317989/648270     |
      | 1276901/15435      -159252/1715        230980/9261     -34061/2205     915331/21609       -2004788/9261     -4479112/77175    -45952316/324135     263698/9261      -1007981/27783   558624/8575       -250571/1029      -1043633/21609      17290246/108045     |
      | 369800/16807       -9229985/403368     94600/7203      79586/7203      0                  -20640/2401       465656/50421      -288/343             -828812/108045   -880/343         -13855246/1260525 -243857/43218     -6050/3087          -7853/13230         |
      | -761272/7203       53511083/432180     -1819739/21609  8515732/77175   10320/2401         -407096/5145      46986416/108045   -42464/2401          -98051056/324135 -334106/15435    -259737284/540225 -2279512/46305    2916997/259308      -32009633/972405    |
      | 39178952/77175     -4672744/8575       13488016/64827  8958224/77175   12522376/84035     -93972832/108045  0                 -56349728/151263     -10743392/108045 -68814832/324135 -512072/77175     -294901328/324135 -1233871888/6806835 856635688/2268945   |
      | 0                  332929/24696        -248110/7203    2553802/15435   369800/16807       -1522544/7203     39178952/77175    -8081672/252105      -24242740/64827  -1456562/21609   -6216358/11025    -22537451/129654  -15249835/1210104   -452838161/13613670 |

               14       14
o31 : Matrix QQ   <-- QQ

i32 : F2 = map(S,S,transpose rhog2);

o32 : RingMap S <-- S

i33 : F2(SpGr36)==SpGr36

o33 = true

i34 : F3 = map(S,S,transpose rhog3);

o34 : RingMap S <-- S

i35 : F3(SpGr36)==SpGr36

o35 = true

We check that moving \(M_{9,1}\) by these elements does not change the value of \(F_{9,1}\).

i36 : F91(transpose(rhog2*transpose(M91)))

o36 = -69066628702801276747468144869007195496549388986880

o36 : QQ

i37 : F91(transpose(rhog3*transpose(M91)))

o37 = -69066628702801276747468144869007195496549388986880

o37 : QQ

Next, we check that moving the point in \(\operatorname{Gr}(9,14)\) corresponding to the cuspidal curve of genus 9 with nonagonal symmetry does not change the value of \(F_{9,1}\).

i38 : LCusp9 = {z_3-(135/2)*z_4,z_9-(1/432)*z_10,z_5-45*z_12,z_6 + 36*z_11,z_0+1/5*z_13};

i39 : F91(matrixFromHyperplaneEquations LCusp9)

      7406920142307
o39 = -------------
            64

o39 : QQ

i40 : F91(transpose(rhog1*transpose(matrixFromHyperplaneEquations LCusp9)))

      7406920142307
o40 = -------------
            64

o40 : QQ

i41 : F91(transpose(rhog2*transpose(matrixFromHyperplaneEquations LCusp9)))

      7406920142307
o41 = -------------
            64

o41 : QQ

i42 : F91(transpose(rhog3*transpose(matrixFromHyperplaneEquations LCusp9)))

      7406920142307
o42 = -------------
            64

o42 : QQ

Testing the invariance of \(F_{9,2}\)

We test the invariance of \(F_{9,2}\). We continue to use the elements \(g_1,g_2,g_3 \in \operatorname{Sp}(6) \) constructed above. We choose a random matrix \(M_{9,2} \in \operatorname{Mat}_{10 \times 14}\) and consider the point in \(\operatorname{Gr}(10,14)\) corresponding to its row space. We also study the point in \(\operatorname{Gr}(10,14)\) corresponding to the tangent developable of genus 9.

i44 : F92(M92)

      309054206607
o44 = ------------
            2

o44 : QQ

i45 : F92(transpose(rhog1*transpose(M92)))

      309054206607
o45 = ------------
            2

o45 : QQ

i46 : F92(transpose(rhog2*transpose(M92)))

      309054206607
o46 = ------------
            2

o46 : QQ

i47 : F92(transpose(rhog3*transpose(M92)))

      309054206607
o47 = ------------
            2

o47 : QQ

i48 : LTD9 = {z_3-(135/2)*z_4,z_9-(1/432)*z_10,z_5-45*z_12,z_6 + 36*z_11};

i49 : F92(matrixFromHyperplaneEquations LTD9)

        1323
o49 = - ----
         16

o49 : QQ

i50 : F92(transpose(rhog1*transpose(matrixFromHyperplaneEquations LTD9)))

        1323
o50 = - ----
         16

o50 : QQ

i51 : F92(transpose(rhog2*transpose(matrixFromHyperplaneEquations LTD9)))

        1323
o51 = - ----
         16

o51 : QQ

i52 : F92(transpose(rhog3*transpose(matrixFromHyperplaneEquations LTD9)))

        1323
o52 = - ----
         16

o52 : QQ