Computer calculations for "Some singular curves in Mukai's model of \(\overline{M}_7\)", Section 6

Code 6.3: Analyzing the irreducible submodules of \( \Lambda^7 S^{+} \)

We compute the dimensions of the irreducible modules in \( \Lambda^7 S^{+} \).

Magma V2.27-8     Fri Apr 14 2023 17:33:00 on MAC-M26AQ05N [Seed = 2362519203]

+-------------------------------------------------------------------+
|       This copy of Magma has been made available through a        |
|                   generous initiative of the                      |
|                                                                   |
|                         Simons Foundation                         |
|                                                                   |
| covering U.S. Colleges, Universities, Nonprofit Research entities,|
|               and their students, faculty, and staff              |
+-------------------------------------------------------------------+

Type ? for help.  Type -D to quit.
> R:=RootDatum("D5" : Isogeny := "SC");
> DecomposeExteriorPower(R,7,[0,0,0,1,0]);
[
    (1 0 1 0 1),
    (3 0 0 1 0)
]
[ 1, 1 ]
> WeylDimension(R,[1,0,1,0,1]);
8800
> WeylDimension(R,[3,0,0,1,0]);
2640

Next, we compute highest weight vectors for the modules \(V_1\) and \(V_2\). The weight spaces of weights \( (1,0,1,0,1)\) and \( (3,0,0,1,0)\) in \( \Lambda^7 S^{+} \) are one-dimensional, and thus, we need only find wedges of the correct weights.

Here are the files LieAlgebraRepresentations.v2.3.m2.txt and SpinRepresentations.v1.7.m2.txt used in the session below.

Macaulay2, version 1.20
with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems, Isomorphism,
               LLLBases, MinimalPrimes, OnlineLookup, PrimaryDecomposition, ReesAlgebra, Saturation,
               TangentCone

i1 : loadPackage("Cyclotomic");

i2 : load "LieAlgebraRepresentations.v2.3.m2";

i3 : load "SpinRepresentations.v1.7.m2";

i4 : n=5;

i5 : B = so2nBasis(n);

i6 : so2nRaisingOperators = so2nPositiveRoots(n);

i7 : PhiminusIndices = select(#B, i -> i >=n and not member(B_i,so2nRaisingOperators));

i8 : sB = spinBasis(n,QQ);

i9 : Bstar = so2nDualBasis(n);

i10 : BstarPerm = flatten apply(#Bstar, i -> flatten select(#B, j -> Bstar_i == 1/(4*(n-1))*B_j));

i11 : splusBDense = apply(sB, M -> M_(apply(2^(n-1), i -> i))^(apply(2^(n-1), i -> i)));

i12 : splusB = apply(splusBDense, M -> sparseRep(M));

i13 : EWB = evenWedgeBasis(n);

i14 : k=7;

i15 : snk = sort subsets(apply(2^(n-1), i -> i),k);

i16 : ZtoW = new HashTable from apply(#snk, i -> {i,snk_i});

i17 : WtoZ = new HashTable from apply(#snk, i -> {snk_i,i});

i18 : ringSubscripts = apply(snk, s -> apply(s, i -> EWB_i));

i19 : K=cyclotomicField(4);

i20 : R=K[apply(ringSubscripts, i -> yy_i),MonomialOrder=>Lex];

i21 : WedgekSplusBasis = apply(splusB, X -> XactionOnWedgek(X,k,ZtoW,WtoZ));

i22 : Splusweights = apply(2^(n-1), j -> apply(n, i -> splusBDense_i_(j,j)));

i23 : Wedgekweights = apply(snk, s -> sum apply(s, j -> LtoD(Splusweights_j)));

i24 : select(#snk, i -> Wedgekweights_i=={1,0,1,0,1})

o24 = {6291}

o24 : List

i25 : R_6291

o25 = yy
        {{1, 2}, {1, 3}, {1, 2, 3, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {1, 3, 4, 5}, {2, 3, 4, 5}}

o25 : R

i26 : select(#snk, i -> Wedgekweights_i=={3,0,0,1,0})

o26 = {5495}

o26 : List

i27 : R_5495

o27 = yy
        {{1, 2}, {1, 3}, {1, 4}, {1, 2, 3, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {1, 3, 4, 5}}

o27 : R