Computer calculations for "Some singular curves in Mukai's model of
\(\overline{M}_7\)", Section 6
Code 6.4: Analyzing \( \operatorname{Sym}^2 V_1 \otimes \operatorname{Sym}^2 V_2 \)
First, we compute the dimension of the \(\operatorname{Spin}(10)\)
invariants in \( \operatorname{Sym}^2 V_1 \otimes \operatorname{Sym}^2 V_2 \).
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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]);
> D1:=LieRepresentationDecomposition(R,[[1,0,1,0,1]],[1]);
> D2:=LieRepresentationDecomposition(R,[[3,0,0,1,0]],[1]);
> X1:=SymmetricPower(D1,2);
> X2:=SymmetricPower(D2,2);
> T:=TensorProduct(X1,X2);
> Multiplicity(T,[0,0,0,0,0]);
89
For any simple Lie group \(G\) and irreducible module \( V(\lambda)\),
we have that \( (V(\lambda) \otimes V(\lambda^{*}))^{G} = 1\).
For each irreducible module \( V(\lambda) \subset \operatorname{Sym}^2
V_1\), we compute the multiplicity of \( V(\lambda) \subset \operatorname{Sym}^2
V_1\) and the multiplicity of \( V(\lambda^*) \subset \operatorname{Sym}^2
V_2\), and show that the sum of these products is 89. Thus, all 89
invariants in \( \operatorname{Sym}^2 V_1 \otimes
\operatorname{Sym}^2 V_2 \) arise in this manner.
> Wts1,Mults1:=WeightsAndMultiplicities(X1);
> Wts2,Mults2:=WeightsAndMultiplicities(X2);
> s:=0;
> for i:=1 to #Wts1 do
for> x:=Wts1[i];
for> xdual:=[x[1],x[2],x[3],x[5],x[4]];
for> m1:=Mults1[i];
for> m2:=Multiplicity(X2,xdual);
for> if m1*m2 ne 0 then
for|if> Sprint(x) cat ": " cat Sprint([m1,m2]);
for|if> s:=s+m1*m2;
for|if> end if;
for> end for;
(3 1 0 0 0): [ 2, 1 ]
(0 1 1 0 0): [ 6, 2 ]
(0 2 0 2 0): [ 4, 1 ]
(3 0 0 1 1): [ 4, 1 ]
(0 2 0 0 2): [ 6, 1 ]
(5 0 0 0 0): [ 1, 1 ]
(1 2 0 0 0): [ 4, 1 ]
(2 0 0 0 2): [ 5, 1 ]
(4 0 0 0 2): [ 2, 1 ]
(4 0 0 2 0): [ 1, 1 ]
(2 2 0 0 2): [ 1, 1 ]
(1 0 0 1 1): [ 5, 1 ]
(1 3 0 0 0): [ 1, 1 ]
(0 0 0 2 0): [ 2, 1 ]
(3 0 0 0 0): [ 2, 1 ]
(0 3 1 0 0): [ 1, 1 ]
(1 2 0 1 1): [ 4, 1 ]
(2 1 1 0 0): [ 5, 2 ]
(1 1 0 0 0): [ 2, 1 ]
(2 0 0 2 0): [ 5, 1 ]
(1 1 0 1 1): [ 10, 1 ]
(0 0 0 0 2): [ 2, 1 ]
(1 0 0 0 0): [ 1, 1 ]
(3 1 0 1 1): [ 1, 1 ]
(3 2 0 0 0): [ 1, 1 ]
> s;
89
