Fordham University

Computing equations of the adjoint variety of \(G_2\)

The homogeneous space \(X_{10}\) is the adjoint variety of \(G_2\). We use an algorithm due to Lichtenstein to compute its equations.

The main theorem in Lichtenstein's 1982 paper says that the equations we seek are obtained by setting \( \operatorname{Cas}(f) = kf\), where \(f\) is a degree 2 tensor, \( \operatorname{Cas}\) is the Casimir operator, and \(k =\langle 2 \lambda + 2 \rho, 2 \lambda\rangle\).

For \( \mathfrak{g}_2\), the adjoint representation has highest weight \( \omega_2\), or \((0,1)\) in the basis \(\{\omega_1,\omega_2\}\). Thus \( \langle 2 \lambda + 2 \rho, 2 \lambda\rangle = \langle 2 (0,1) + 2 (1,1), 2 (0,1)\rangle\). In the Macaualay2 session below, we compute this value to be 20. 

Macaulay2, version 1.26.05
Type "help" to see useful commands

i1 : needsPackage("LieAlgebraRepresentations");

i2 : g2 = simpleLieAlgebra("G",2);

i3 : V01 = irreducibleLieAlgebraModule(g2,{0,1});

i4 : killingForm(g2,2*{0,1}+2*{1,1},2*{0,1})

o4 = 20

o4 : QQ
Next we construct the Casimir operator on the tensor square of the adjoint representation. 
i5 : rhoV01 = adjointRepresentation(g2);

i6 : rhoT2V01 = rhoV01**rhoV01;

i7 : Cas = casimirOperator(rhoT2V01);

              196       196
o7 : Matrix QQ    <-- QQ

i8 : Cas==transpose Cas

o8 = false
We compute the ideal of equations of the form \( \operatorname{Cas}(f) = kf\). 
i9 : Id = matrix apply(numrows Cas, i -> apply(numrows Cas, j -> if i==j then 1 else 0/1));

              196       196
o9 : Matrix QQ    <-- QQ

i10 : LAB = rhoV01#"Basis";

i11 : S= ring(first(LAB#"BasisElements"))

o11 = G2ring

o11 : PolynomialRing

i12 : B2 = flatten apply(14, i -> apply(14, j -> S_i*S_j));

i13 : I = ideal((Cas-20*Id)*(transpose matrix {B2}));

o13 : Ideal of G2ring

i14 : L = flatten entries mingens I;

i15 : print toString(L)<<endl
{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}

Next, we test that these equations are invariant under some elements of \(G_2\), and that they vanish on the orbit of the highest weight vector.

We load a useful function and change the coefficient ring.

i16 : eval = (p,f) -> (
          R:=ring(f);
          CR:=coefficientRing(R);
          lift(substitute(f, apply(numgens R, i -> R_i=>p_i)),CR)
      );

i17 : L = rhoV01#"RepresentationMatrices";

i18 : R=frac(QQ[c,t,Degrees=>{0,0}]);

i19 : 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];

i20 : 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};

o20 : Ideal of S
We construct some elements of \(G_2\). See TestInvarianceGenus10.htm for more discussion. 
i21 : G2elementsa = apply(2, i -> diagonalMatrix(apply(14, j ->t^(lift((L_i)_(j,j),ZZ)))));

i22 : truncatedMatrixExponential = (n,M) -> sum apply(n, i -> M^i/(i!));

i23 : G2elementsb = apply(toList(2..13), i -> truncatedMatrixExponential(4,c*(L_i)));

i24 : G2elements = join(G2elementsa,G2elementsb);
We check that the ideal of the \(G_2\) adjoint variety is invariant under the actions of these elements. 
i25 : G2elementmaps = apply(G2elements, g -> map(S,S,transpose g));

i26 : all(G2elementmaps, F -> F(G2AdjVar)==G2AdjVar)

o26 = true
The highest weight vector in the adjoint representation is \(X_6\). We check that the quadrics we computed vanish on this point, and on its orbit under the elements of \(G_2\) we computed. 
i27 : p = apply(14, i -> if i==7 then 1 else 0);

i28 : all(flatten entries gens G2AdjVar, f -> eval(p,f)==0)

o28 = true

i29 : gps = apply(G2elements, g -> flatten entries(g*(transpose matrix {p})));

i30 : all(gps, x -> all(flatten entries gens G2AdjVar, f -> eval(x,f)==0))

o30 = true
Some of the elements of \(G_2\) used above fix the point \(p\). For an additional test, we construct a random element of \(G_2\) and test that our quadrics vanish at the image of the point under this element. 
i31 : c = {-6, 5, -1, -9, 9, 4, -7, 8, 10, -4, 6, -5, -10, -8};

i32 : G2elementsa = apply(2, i -> diagonalMatrix(apply(14, j ->(c_i)^(lift((L_i)_(j,j),ZZ)))));

i33 : G2elementsb = apply(toList(2..13), i -> truncatedMatrixExponential(4,c_i*(L_i)));

i34 : G2elements = join(G2elementsa,G2elementsb);

i35 : g = product G2elements

o35 = | -60647       47088       -102555      133294       -532649      686846       341125      238302       -12678     -5630        -1057      -2710      577         -145      |
      | -87720       67325       -146547      188068       -750207      966051       480892      338981       -17469     -7629        -438       -2988      613         -155      |
      | -360036      1258956/5   -547488      2939832/5    -12099492/5  3271176      8490816/5   4955256/5    -28872     -124992/5    53388/5    3060       -4284/5     1044/5    |
      | -7510325/36  3890525/27  -2821075/9   71255425/216 -97683125/72 132105925/72 68849975/72 60359675/108 -177725/12 -2855725/216 639475/72  290675/72  -223675/216 1525/6    |
      | 1281140      -1728495/2  5636200/3    -3752735/2   23338255/3   -63934235/6  -16864960/3 -6265255/2   57155      223520/3     -408365/6  -118825/3  19385/2     -4785/2   |
      | -8841468     5812536     -12624942    11870922     -49642290    68943060     36885048    19540026     -155130    -459336      581820     390474     -93522      23142     |
      | -226371672/5 136743984/5 -296436024/5 217838376/5  -189976104   1403677512/5 794603952/5 330623208/5  15174864/5 -7456536/5   4738176    19236528/5 -4492368/5  1114992/5 |
      | -55985430    40294385    -87674280    100634695    -407638275   538541715    275067305   176245935    -6772770   -3995345     1492725    27285      -44245      9715      |
      | -65365/18    2250        -175625/36   70705/18     -33515/2     432955/18    120025/9    226175/36    5045/36    -1175/9      2155/6     835/3      -590/9      65/4      |
      | 1967976/25   -1251072/25 2714472/25   -2326752/25  9902088/25   -14107608/25 -7711632/25 -3691224/25  -38232/25  90504/25     -152928/25 -114696/25 1080        -6696/25  |
      | 62556/5      -41592/5    18066        -17424       363378/5     -502332/5    -53448      -143886/5    1746/5     696          -3636/5    -2316/5    558/5       -138/5    |
      | 11765/6      -4003/3     2900         -8846/3      73235/6      -100015/6    -8760       -29593/6     619/6      120          -275/3     -295/6     73/6        -3        |
      | -11125/36    23395/108   -33925/72    13775/27     -150665/72   101375/36    315025/216  187345/216   -635/24    -2275/108    725/72     55/18      -185/216    5/24      |
      | 6            -48/5       21           -248/5       924/5        -210         -91         -504/5       48/5       2            3          18/5       -4/5        1/5       |

               14       14
o35 : Matrix QQ   <-- QQ

i36 : gp =flatten entries(g*(transpose matrix {p}))

                       4955256  60359675    6265255            330623208             226175    3691224    143886    29593  187345    504
o36 = {238302, 338981, -------, --------, - -------, 19540026, ---------, 176245935, ------, - -------, - ------, - -----, ------, - ---}
                          5        108         2                   5                   36         25         5        6      216      5

o36 : List

i37 : all(flatten entries gens G2AdjVar, f -> eval(gp,f)==0)

o37 = true