Computer calculations for "Some singular curves in Mukai's model of
\(\overline{M}_7\)", Section 6
Code 6.5: Dual pairs in \( \operatorname{Sym}^2 V_1 \otimes \operatorname{Sym}^2 V_2 \)
We study the irreducible representations \( V(\lambda) \subset \operatorname{Sym}^2
V_1\) whose dual \( V(\lambda^*) \) is a summand in \( \operatorname{Sym}^2
V_2\).
For each dual pair, we compute three dimensions:
\begin{eqnarray}
d(\lambda) & = & \dim V(\lambda) \\
d_1(\lambda) & = & \dim (\operatorname{Sym}^2 V_1)_{\lambda} \\
d_2(\lambda) & = & \dim (\operatorname{Sym}^2 V_2)_{\lambda^*} \\
\end{eqnarray}
as well as \(\varphi(\lambda) = \max \{ d(\lambda) d_1(\lambda) , d(\lambda) d_2(\lambda) \}\).
Magma V2.27-8 Sat Apr 15 2023 13:51:14 on MAC-M26AQ05N [Seed = 2429423849]
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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);
> Wts1,Mults1:=WeightsAndMultiplicities(X1);
> Wts2,Mults2:=WeightsAndMultiplicities(X2);
> dimVlambdasubmu:=function(R,lambda,mu)
function> W,M:=DominantWeights(R,lambda);
function> for i:=1 to #W do
function|for> if [W[i][k]: k in [1..Rank(R)]] eq [mu[k]: k in [1..Rank(R)]] the\
n
function|for|if> return M[i];
function|for|if> end if;
function|for> end for;
function> return 0;
function> end function;
>
>
> dimWmu:=function(R,D,mu)
function> s:=0;
function> W,M:=WeightsAndMultiplicities(D);
function> for i:=1 to #W do
function|for> s:=s+M[i]*dimVlambdasubmu(R,W[i],mu);
function|for> end for;
function> return s;
function> end function;
> 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> d:=WeylDimension(R,x);
for|if> d1:=dimWmu(R,X1,x);
for|if> d2:=dimWmu(R,X2,xdual);
for|if> m:=Maximum(d1,d2);
for|if> Sprint(x) cat ": " cat Sprint([d,d1,d2,d*m]);
for|if> s := s + m1*m2;
for|if> end if;
for> end for;
(3 1 0 0 0): [ 4608, 2204, 448, 10156032 ]
(0 1 1 0 0): [ 2970, 14946, 1326, 44389620 ]
(0 2 0 2 0): [ 46800, 818, 66, 38282400 ]
(3 0 0 1 1): [ 28160, 1356, 230, 38184960 ]
(0 2 0 0 2): [ 46800, 909, 70, 42541200 ]
(5 0 0 0 0): [ 1782, 232, 160, 413424 ]
(1 2 0 0 0): [ 4410, 5984, 672, 26389440 ]
(2 0 0 0 2): [ 4950, 6108, 590, 30234600 ]
(4 0 0 0 2): [ 50050, 84, 32, 4204200 ]
(4 0 0 2 0): [ 50050, 72, 32, 3603600 ]
(2 2 0 0 2): [ 831600, 6, 2, 4989600 ]
(1 0 0 1 1): [ 1728, 23099, 2029, 39915072 ]
(1 3 0 0 0): [ 37632, 264, 52, 9934848 ]
(0 0 0 2 0): [ 126, 35260, 3040, 4442760 ]
(3 0 0 0 0): [ 210, 14704, 1816, 3087840 ]
(0 3 1 0 0): [ 258720, 20, 4, 5174400 ]
(1 2 0 1 1): [ 380160, 158, 19, 60065280 ]
(2 1 1 0 0): [ 68640, 814, 128, 55872960 ]
(1 1 0 0 0): [ 320, 35236, 3352, 11275520 ]
(2 0 0 2 0): [ 4950, 5968, 580, 29541600 ]
(1 1 0 1 1): [ 36750, 3784, 348, 139062000 ]
(0 0 0 0 2): [ 126, 35525, 3040, 4476150 ]
(1 0 0 0 0): [ 10, 80368, 7600, 803680 ]
(3 1 0 1 1): [ 436590, 42, 13, 18336780 ]
(3 2 0 0 0): [ 48510, 72, 36, 3492720 ]
> s;
89
We see that \( \lambda = (5,0,0,0,0)\) minimizes \( \varphi(\lambda)\).
