First we create a matrix with the evaluations of the 147 covariants from Tables
5.4and5.5at 2750 random integers.
Next we show that the 147 covariants generate C(V
7)
d,mwith d ≤ 23 and
m ≤ 15. For each d ≤ 24 and m ≤ 15 we do the following: we construct all
possible monomials of degree d and order m spanned by the 147 covariants and
evaluate them at random integers (using the matrix we just created). These
monomials will span C(V
7)
d,mif their evaluation matrix has the rank equal to
called mon{deg,ord} which returns a list of all covariants degree deg and order
ord as monomials of degree deg spanned by the above 147 covariants.
In[]:=listdeg={{1,7},{2,2},{2,6},{2,10},{3,3},{3,5},{3,7},{3,9},{3,11}, {3,15},{4,0},{4,4},{4,4},{4,6},{4,8},{4,8},{4,10},{4,14},{5,1},{5,3}, {5,3},{5,5},{5,5},{5,7},{5,7},{5,9},{5,9},{5,13},{6,2},{6,2},{6,2}, {6,4},{6,4},{6,6},{6,6},{6,8},{6,8},{6,12},{7,1},{7,1},{7,1},{7,3}, {7,3},{7,5},{7,5},{7,5},{7,5},{7,7},{7,7},{7,11},{8,0},{8,0},{8,0}, {8,2},{8,2},{8,2},{8,4},{8,4},{8,4},{8,6},{8,6},{8,6},{8,10},{9,1}, {9,1},{9,1},{9,3},{9,3},{9,3},{9,3},{9,3},{9,5},{9,5},{9,9},{10,2}, {10,2},{10,2},{10,2},{10,4},{10,4},{10,4},{10,4},{10,8},{11,1}, {11,1},{11,1},{11,1},{11,1},{11,3},{11,3},{11,3},{11,7},{12,0},{12,0}, {12,0},{12,0},{12,0},{12,0},{12,2},{12,2},{12,2},{12,2},{12,2},{12,2}, {12,6},{13,1},{13,1},{13,1},{13,1},{13,1},{13,1},{13,1},{13,3},{13,5}, {14,0},{14,0},{14,0},{14,0},{14,4},{14,4},{15,1},{15,1},{15,1},{15,3}, {16,0},{16,0},{16,2},{16,2},{16,2},{17,1},{17,1},{18,0},{18,0},{18,0}, {18,0},{18,0},{18,0},{18,0},{18,0},{18,0},{19,1},{20,0},{22,0},{22,0}, {23,1},{26,0},{30,0}}; In[]:=listgen={FF,LL,CHI,HH,RR,C35,C37,Epsilon,Gamm,TT,Inv4,PP,C44,C46, Delta,C48,C410,Zet,C51,C531,C532,GG,C55,ETA,C57,C591,C592,C513,Tau, C621,C622,C641,C642,C661,C662,Bet,C68,C612,Alpha,C711,C712,C731,C732, Theta,C751,C752,C753,C771,C772,Cov711,C801,C802,C803,C821,C822,C823,MU, C841,C842,Lambda,C861,C862,C810,C911,C912,C913,QQ,C931,C932,C933,C934, C951,C952,C99,NU,C1021,C1022,C1023,C1041,C1042,C1043,C1044,C108,C1111, C1112,C1113,C1114,C1115,PHI,C1131,C1132,C117,Rbig,C1201,C1202,C1203, C1204,C1205,C1221,C1222,C1223,C1224,C1225,C1226,C126,RO,C1311,C1312, C1313,C1314,C1315,C1316,C133,C135,C1401,C1402,C1403,C1404,PSI,C1441, C1511,C1512,C1513,C153,C1601,C1602,Sigma,C1621,C1622,Omega,C1711,C1801, C1802,C1803,C1804,C1805,C1806,C1807,C1808,C1809,C191,C201,C2201,C2202, C231,C260,C300}; In[]:=mon[md_,j_]:= mon[md,j]=If[md=={0,0},1,If[md[[1]]<0||md[[2]]<0,{}, Flatten[Table[listgen[[i]]mon[md-listdeg[[i]],i],{i,1,j}]]]]
Here we show that the 147 covariants generate C(V
7)
d,mwith d ≤ 24 and
m ≤ 15:
In[]:= Timing[ For[deg=2,deg<24,deg++,ord=1; While[ord<16, symm=SymTensor[7,deg]/.Table[v[i]->0,{i,16,1000}]; kk=Coefficient[symm,v[ord]]+1; If[kk!=1,Print[{deg,ord},": the vector space of covs has dimension",kk-1]; ss=mon[{deg,ord},147];
If[ss!={},
tt=Intersection[ss,listgen]; If[tt!={},
ss1=Complement[ss,tt];
eval1 = Table[ss1 /.Table[listgen[[j]]->Matr[[k]][[j]][[1]], {j,1,Length[Matr[[k]]]}],{k,1,kk}];
rk1=If[ss1=={},0,MatrixRank[eval1,Modulus->32003]]; Print["-- the subpace spanned by covs of degree < ", deg,
" has dim >=", rk1];
Print["-- there are ", Length[tt], " gens of degree ", deg]; eval=Table[ss/.Table[listgen[[j]]->Matr[[k]][[j]][[1]],
rk=MatrixRank[eval,Modulus->32003];
Print["-- the subpace spanned by covs of degree <= ", deg, " has dim >=", rk], If[Length[ss]>3000,DIM=kk-1; dd1=RandomSample[ss,kk+10]; dd2=Complement[ss,dd1]; eval=Table[dd1/.Table[listgen[[j]]->Matr[[k]][[j]][[1]], {j,1,Length[Matr[[k]]]}],{k,1,kk}]; eval1=ColumnEchelon[eval]; rk1=MatrixRank[eval1,Modulus->32003]; rk=rk1; While[rk!=DIM, AA=Table[RandomInteger[{-10,10}],{i,1,50}]; uu=RandomSample[dd2,50].AA; evaluu=Table[uu/.Table[listgen[[j]]->Matr[[k]][[j]][[1]], {j,1,Length[Matr[[k]]]}],{k,1,kk}]; eval=Table[Join[eval1[[i]],{evaluu[[i]]}],{i,1,kk}]; eval=Mod[eval,32003]; rk=MatrixRank[eval,Modulus->32003]; If[rk>rk1,eval1=eval;rk1=rk]];
Print["-- the subpace spanned by covs of degree < ", deg, " has dim >=", rk],
eval=Table[ss/.Table[listgen[[j]]->Matr[[k]][[j]][[1]], {j,1,Length[Matr[[k]]]}],{k,1,kk}];
rk=MatrixRank[eval,Modulus->32003];
Print["-- the subpace spanned by covs of degree < ", deg, " has dim >=", rk]]]]]; ord++]
]]
{2,2}: the vector space of covs has dimension 1
-- the subpace spanned by covs of degree < 2 has dim >=0 -- there are 1 gens of degree 2
-- the subpace spanned by covs of degree <= 2 has dim >=1 {2,6}: the vector space of covs has dimension 1
-- the subpace spanned by covs of degree < 2 has dim >=0 -- there are 1 gens of degree 2
-- the subpace spanned by covs of degree <= 2 has dim >=1 {2,10}: the vector space of covs has dimension 1
-- the subpace spanned by covs of degree < 2 has dim >=0 -- there are 1 gens of degree 2
-- the subpace spanned by covs of degree <= 2 has dim >=1 {2,14}: the vector space of covs has dimension 1
-- the subpace spanned by covs of degree < 2 has dim >=1 {3,3}: the vector space of covs has dimension 1
-- the subpace spanned by covs of degree < 3 has dim >=0 -- there are 1 gens of degree 3
-- the subpace spanned by covs of degree <= 3 has dim >=1 {3,5}: the vector space of covs has dimension 1
-- the subpace spanned by covs of degree < 3 has dim >=0 -- there are 1 gens of degree 3
-- the subpace spanned by covs of degree <= 3 has dim >=1 {3,7}: the vector space of covs has dimension 1
-- the subpace spanned by covs of degree < 3 has dim >=0 -- there are 1 gens of degree 3
-- the subpace spanned by covs of degree <= 3 has dim >=1 {3,9}: the vector space of covs has dimension 2
-- the subpace spanned by covs of degree < 3 has dim >=1 -- there are 1 gens of degree 3
-- the subpace spanned by covs of degree <= 3 has dim >=2 {3,11}: the vector space of covs has dimension 1
-- the subpace spanned by covs of degree < 3 has dim >=0 -- there are 1 gens of degree 3
-- the subpace spanned by covs of degree <= 3 has dim >=1 {3,13}: the vector space of covs has dimension 1
-- the subpace spanned by covs of degree < 3 has dim >=1 {3,15}: the vector space of covs has dimension 1
-- the subpace spanned by covs of degree < 3 has dim >=0 -- there are 1 gens of degree 3
-- the subpace spanned by covs of degree <= 3 has dim >=1 {4,4}: the vector space of covs has dimension 3
-- the subpace spanned by covs of degree < 4 has dim >=1 -- there are 2 gens of degree 4
-- the subpace spanned by covs of degree <= 4 has dim >=3 {4,6}: the vector space of covs has dimension 1
-- the subpace spanned by covs of degree < 4 has dim >=0 -- there are 1 gens of degree 4
-- the subpace spanned by covs of degree <= 4 has dim >=1 {4,8}: the vector space of covs has dimension 3
-- the subpace spanned by covs of degree < 4 has dim >=1 -- there are 2 gens of degree 4
-- the subpace spanned by covs of degree <= 4 has dim >=3 {4,10}: the vector space of covs has dimension 2
-- the subpace spanned by covs of degree < 4 has dim >=1 -- there are 1 gens of degree 4
-- the subpace spanned by covs of degree <= 4 has dim >=2 {4,12}: the vector space of covs has dimension 3
-- the subpace spanned by covs of degree < 4 has dim >=3 {4,14}: the vector space of covs has dimension 2
-- the subpace spanned by covs of degree < 4 has dim >=1 -- there are 1 gens of degree 4
-- the subpace spanned by covs of degree <= 4 has dim >=2 {5,1}: the vector space of covs has dimension 1
-- the subpace spanned by covs of degree < 5 has dim >=0 -- there are 1 gens of degree 5
-- the subpace spanned by covs of degree <= 5 has dim >=1 {5,3}: the vector space of covs has dimension 2
-- the subpace spanned by covs of degree < 5 has dim >=0 -- there are 2 gens of degree 5
-- the subpace spanned by covs of degree <= 5 has dim >=2 {5,5}: the vector space of covs has dimension 3
-- the subpace spanned by covs of degree < 5 has dim >=1 -- there are 2 gens of degree 5
-- the subpace spanned by covs of degree <= 5 has dim >=3 {5,7}: the vector space of covs has dimension 4
-- the subpace spanned by covs of degree < 5 has dim >=2 -- there are 2 gens of degree 5
-- the subpace spanned by covs of degree <= 5 has dim >=4 {5,9}: the vector space of covs has dimension 4
-- the subpace spanned by covs of degree < 5 has dim >=2 -- there are 2 gens of degree 5
-- the subpace spanned by covs of degree <= 5 has dim >=4 {5,11}: the vector space of covs has dimension 5
-- the subpace spanned by covs of degree < 5 has dim >=5 {5,13}: the vector space of covs has dimension 4
-- there are 1 gens of degree 5
-- the subpace spanned by covs of degree <= 5 has dim >=4 {5,15}: the vector space of covs has dimension 5
-- the subpace spanned by covs of degree < 5 has dim >=5 {6,2}: the vector space of covs has dimension 4
-- the subpace spanned by covs of degree < 6 has dim >=1 -- there are 3 gens of degree 6
-- the subpace spanned by covs of degree <= 6 has dim >=4 {6,4}: the vector space of covs has dimension 2
-- the subpace spanned by covs of degree < 6 has dim >=0 -- there are 2 gens of degree 6
-- the subpace spanned by covs of degree <= 6 has dim >=2 {6,6}: the vector space of covs has dimension 7
-- the subpace spanned by covs of degree < 6 has dim >=5 -- there are 2 gens of degree 6
-- the subpace spanned by covs of degree <= 6 has dim >=7 {6,8}: the vector space of covs has dimension 5
-- the subpace spanned by covs of degree < 6 has dim >=3 -- there are 2 gens of degree 6
-- the subpace spanned by covs of degree <= 6 has dim >=5 {6,10}: the vector space of covs has dimension 8
-- the subpace spanned by covs of degree < 6 has dim >=8 {6,12}: the vector space of covs has dimension 7
-- the subpace spanned by covs of degree < 6 has dim >=6 -- there are 1 gens of degree 6
-- the subpace spanned by covs of degree <= 6 has dim >=7 {6,14}: the vector space of covs has dimension 9
-- the subpace spanned by covs of degree < 6 has dim >=9 {7,1}: the vector space of covs has dimension 3
-- the subpace spanned by covs of degree < 7 has dim >=0 -- there are 3 gens of degree 7
-- the subpace spanned by covs of degree <= 7 has dim >=3 {7,3}: the vector space of covs has dimension 4
-- the subpace spanned by covs of degree < 7 has dim >=2 -- there are 2 gens of degree 7
-- the subpace spanned by covs of degree <= 7 has dim >=4 {7,5}: the vector space of covs has dimension 7
-- the subpace spanned by covs of degree < 7 has dim >=3 -- there are 4 gens of degree 7
-- the subpace spanned by covs of degree <= 7 has dim >=7 {7,7}: the vector space of covs has dimension 9
-- the subpace spanned by covs of degree < 7 has dim >=7 -- there are 2 gens of degree 7
-- the subpace spanned by covs of degree <= 7 has dim >=9 {7,9}: the vector space of covs has dimension 10
-- the subpace spanned by covs of degree < 7 has dim >=10 {7,11}: the vector space of covs has dimension 11
-- the subpace spanned by covs of degree < 7 has dim >=10 -- there are 1 gens of degree 7
-- the subpace spanned by covs of degree <= 7 has dim >=11 {7,13}: the vector space of covs has dimension 13
-- the subpace spanned by covs of degree < 7 has dim >=13 {7,15}: the vector space of covs has dimension 12
-- the subpace spanned by covs of degree < 7 has dim >=12 {8,2}: the vector space of covs has dimension 3
-- the subpace spanned by covs of degree < 8 has dim >=0 -- there are 3 gens of degree 8
-- the subpace spanned by covs of degree <= 8 has dim >=3 {8,4}: the vector space of covs has dimension 10
-- the subpace spanned by covs of degree < 8 has dim >=7 -- there are 3 gens of degree 8
-- the subpace spanned by covs of degree <= 8 has dim >=10 {8,6}: the vector space of covs has dimension 9
-- the subpace spanned by covs of degree < 8 has dim >=6 -- there are 3 gens of degree 8
-- the subpace spanned by covs of degree <= 8 has dim >=9 {8,8}: the vector space of covs has dimension 16
-- the subpace spanned by covs of degree < 8 has dim >=16 {8,10}: the vector space of covs has dimension 14
-- the subpace spanned by covs of degree < 8 has dim >=13 -- there are 1 gens of degree 8
-- the subpace spanned by covs of degree <= 8 has dim >=14 {8,12}: the vector space of covs has dimension 19
-- the subpace spanned by covs of degree < 8 has dim >=19 {8,14}: the vector space of covs has dimension 17
-- the subpace spanned by covs of degree < 8 has dim >=17 {9,1}: the vector space of covs has dimension 4
-- the subpace spanned by covs of degree < 9 has dim >=1 -- there are 3 gens of degree 9
-- the subpace spanned by covs of degree <= 9 has dim >=4 {9,3}: the vector space of covs has dimension 10
-- the subpace spanned by covs of degree < 9 has dim >=5 -- there are 5 gens of degree 9
-- the subpace spanned by covs of degree <= 9 has dim >=10 {9,5}: the vector space of covs has dimension 13
-- the subpace spanned by covs of degree < 9 has dim >=11 -- there are 2 gens of degree 9
-- the subpace spanned by covs of degree <= 9 has dim >=13 {9,7}: the vector space of covs has dimension 17
-- the subpace spanned by covs of degree < 9 has dim >=17 {9,9}: the vector space of covs has dimension 21
-- the subpace spanned by covs of degree < 9 has dim >=20 -- there are 1 gens of degree 9
-- the subpace spanned by covs of degree <= 9 has dim >=21 {9,11}: the vector space of covs has dimension 24
-- the subpace spanned by covs of degree < 9 has dim >=24 {9,13}: the vector space of covs has dimension 25
-- the subpace spanned by covs of degree < 9 has dim >=25 {9,15}: the vector space of covs has dimension 29
-- the subpace spanned by covs of degree < 9 has dim >=29 {10,2}: the vector space of covs has dimension 12
-- the subpace spanned by covs of degree < 10 has dim >=8 -- there are 4 gens of degree 10
-- the subpace spanned by covs of degree <= 10 has dim >=12 {10,4}: the vector space of covs has dimension 13
-- the subpace spanned by covs of degree < 10 has dim >=9 -- there are 4 gens of degree 10
-- the subpace spanned by covs of degree <= 10 has dim >=13 {10,6}: the vector space of covs has dimension 23
-- the subpace spanned by covs of degree < 10 has dim >=23 {10,8}: the vector space of covs has dimension 23
-- the subpace spanned by covs of degree < 10 has dim >=22 -- there are 1 gens of degree 10
{10,10}: the vector space of covs has dimension 34
-- the subpace spanned by covs of degree < 10 has dim >=34 {10,12}: the vector space of covs has dimension 31
-- the subpace spanned by covs of degree < 10 has dim >=31 {10,14}: the vector space of covs has dimension 40
-- the subpace spanned by covs of degree < 10 has dim >=40 {11,1}: the vector space of covs has dimension 8
-- the subpace spanned by covs of degree < 11 has dim >=3 -- there are 5 gens of degree 11
-- the subpace spanned by covs of degree <= 11 has dim >=8 {11,3}: the vector space of covs has dimension 16
-- the subpace spanned by covs of degree < 11 has dim >=13 -- there are 3 gens of degree 11
-- the subpace spanned by covs of degree <= 11 has dim >=16 {11,5}: the vector space of covs has dimension 24
-- the subpace spanned by covs of degree < 11 has dim >=24 {11,7}: the vector space of covs has dimension 31
-- the subpace spanned by covs of degree < 11 has dim >=30 -- there are 1 gens of degree 11
-- the subpace spanned by covs of degree <= 11 has dim >=31 {11,9}: the vector space of covs has dimension 38
-- the subpace spanned by covs of degree < 11 has dim >=38 {11,11}: the vector space of covs has dimension 43
-- the subpace spanned by covs of degree < 11 has dim >=43 {11,13}: the vector space of covs has dimension 49
-- the subpace spanned by covs of degree < 11 has dim >=49 {11,15}: the vector space of covs has dimension 52
-- the subpace spanned by covs of degree < 11 has dim >=52 {12,2}: the vector space of covs has dimension 12
-- the subpace spanned by covs of degree < 12 has dim >=6 -- there are 6 gens of degree 12
-- the subpace spanned by covs of degree <= 12 has dim >=12 {12,4}: the vector space of covs has dimension 29
-- the subpace spanned by covs of degree < 12 has dim >=29 {12,6}: the vector space of covs has dimension 33
-- the subpace spanned by covs of degree < 12 has dim >=32 -- there are 1 gens of degree 12
-- the subpace spanned by covs of degree <= 12 has dim >=33 {12,8}: the vector space of covs has dimension 48
-- the subpace spanned by covs of degree < 12 has dim >=48 {12,10}: the vector space of covs has dimension 49
-- the subpace spanned by covs of degree < 12 has dim >=49 {12,12}: the vector space of covs has dimension 65
-- the subpace spanned by covs of degree < 12 has dim >=65 {12,14}: the vector space of covs has dimension 64
-- the subpace spanned by covs of degree < 12 has dim >=64 {13,1}: the vector space of covs has dimension 14
-- the subpace spanned by covs of degree < 13 has dim >=7 -- there are 7 gens of degree 13
-- the subpace spanned by covs of degree <= 13 has dim >=14 {13,3}: the vector space of covs has dimension 26
-- the subpace spanned by covs of degree < 13 has dim >=25 -- there are 1 gens of degree 13
-- the subpace spanned by covs of degree <= 13 has dim >=26 {13,5}: the vector space of covs has dimension 39
-- the subpace spanned by covs of degree < 13 has dim >=38 -- there are 1 gens of degree 13
-- the subpace spanned by covs of degree <= 13 has dim >=39 {13,7}: the vector space of covs has dimension 53
-- the subpace spanned by covs of degree < 13 has dim >=53 {13,9}: the vector space of covs has dimension 62
-- the subpace spanned by covs of degree < 13 has dim >=62 {13,11}: the vector space of covs has dimension 74
-- the subpace spanned by covs of degree < 13 has dim >=74 {13,13}: the vector space of covs has dimension 83
-- the subpace spanned by covs of degree < 13 has dim >=83 {13,15}: the vector space of covs has dimension 91
-- the subpace spanned by covs of degree < 13 has dim >=91 {14,2}: the vector space of covs has dimension 30
-- the subpace spanned by covs of degree < 14 has dim >=30 {14,4}: the vector space of covs has dimension 37
-- the subpace spanned by covs of degree < 14 has dim >=35 -- there are 2 gens of degree 14
-- the subpace spanned by covs of degree <= 14 has dim >=37 {14,6}: the vector space of covs has dimension 62
-- the subpace spanned by covs of degree < 14 has dim >=62 {14,8}: the vector space of covs has dimension 68
-- the subpace spanned by covs of degree < 14 has dim >=68 {14,10}: the vector space of covs has dimension 91
-- the subpace spanned by covs of degree < 14 has dim >=91 {14,12}: the vector space of covs has dimension 95
-- the subpace spanned by covs of degree < 14 has dim >=95 {14,14}: the vector space of covs has dimension 116
-- the subpace spanned by covs of degree < 14 has dim >=116 {15,1}: the vector space of covs has dimension 20
-- the subpace spanned by covs of degree < 15 has dim >=17 -- there are 3 gens of degree 15
-- the subpace spanned by covs of degree <= 15 has dim >=20 {15,3}: the vector space of covs has dimension 42
-- the subpace spanned by covs of degree < 15 has dim >=41 -- there are 1 gens of degree 15
-- the subpace spanned by covs of degree <= 15 has dim >=42 {15,5}: the vector space of covs has dimension 62
-- the subpace spanned by covs of degree < 15 has dim >=62 {15,7}: the vector space of covs has dimension 80
-- the subpace spanned by covs of degree < 15 has dim >=80