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

Code 6.1: \( \operatorname{Spin}(10)\) invariant polynomials in \( \operatorname{Sym}^{\bullet} \Lambda^7 S^{+}\)

By [18 Prop. 20.15 and section 19.2], the representation \(S^{+}\) is irreducible with highest weight \(\omega_4\). The character calculation below shows that there are no \( \operatorname{Spin}(10) \) invariants in \( \operatorname{Sym}^{d} \Lambda^7 S^{+} \) for \(d = 1,2,3\), but there are \( \operatorname{Spin}(10)\) invariants when \(d=4\).

Magma V2.27-8     Fri Apr 14 2023 16:46:28 on MAC-M26AQ05N [Seed = 578813376]

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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 ]
> D:=LieRepresentationDecomposition(R,[[1,0,1,0,1],[3,0,0,1,0]],[1,1]);
> Multiplicity(SymmetricPower(D,1),[0,0,0,0,0]);
0
> Multiplicity(SymmetricPower(D,2),[0,0,0,0,0]);
0
> Multiplicity(SymmetricPower(D,3),[0,0,0,0,0]);
0
> Multiplicity(SymmetricPower(D,4),[0,0,0,0,0]);
392