CAlgRank5A2FromList:=function(L)
local B0,B1,B2,B3,B4,T0,T1,T2,T3,T4,T5,T6,T7,T8,T9,T10,T11,T12, T13,T14,T15,T16,T;

B0:=IdentityMat(5);

B1:=[[ 0, 0, L[17], 0, 0], 
     [1, L[1], L[17]- L[1] -L[5] -L[9] -1, L[5], L[9]], 
     [0, L[2], L[17]- L[2] -L[6] -L[10], L[6], L[10]], 
     [0, L[3], L[17]- L[3] -L[7] -L[11], L[7], L[11]], 
     [0, L[4], L[17]- L[4]-L[8]-L[12], L[8], L[12]]];

B2:=[[ 0, L[17], 0, 0, 0], 
     [0, L[17]- L[2] -L[6] -L[10], L[2], L[10], L[6]], 
     [1, L[17]- L[1] -L[5] -L[9]-1, L[1], L[9], L[5]], 
     [0, L[17]- L[4] -L[8] -L[12], L[4], L[12], L[8]], 
     [0, L[17]- L[3]-L[7]-L[11], L[3], L[11], L[7]]];

B3:=[[ 0, 0, 0, 0, L[18]], 
     [0, L[5], L[10], L[13], L[18]-L[5] -L[10] -L[13]], 
     [0, L[6], L[9], L[14], L[18]-L[6] -L[9] -L[14]], 
     [1, L[7], L[12], L[15], L[18]-L[7] -L[12] -L[15]-1], 
     [0, L[8], L[11], L[16], L[18]-L[8] -L[11] -L[16]]];

B4:=[[ 0, 0, 0, L[18], 0], 
     [0, L[9], L[6], L[18]-L[6] -L[9] -L[14], L[14]], 
     [0, L[10], L[5], L[18]-L[5] -L[10] -L[13], L[13]], 
     [0, L[11], L[8], L[18]-L[8] -L[11] -L[16], L[16]], 
     [1, L[12], L[7], L[18]-L[7] -L[12] -L[15]-1, L[15]]];

T1:=B1^2- (L[1]*B1+L[2]*B2+L[3]*B3+L[4]*B4);
T1:=Concatenation(T1);
T2:=B1*B2- (L[17]*B0+(L[17]- L[1] -L[5] -L[9]-1)*B1+(L[17]- L[2] -L[6] -L[10])*B2+(L[17]- L[3]-L[7]-L[11])*B3+(L[17]- L[4] -L[8] -L[12])*B4);
T2:=Concatenation(T2);
T3:=B1*B3- (L[5]*B1+L[6]*B2+L[7]*B3+L[8]*B4);
T3:=Concatenation(T3);
T4:=B1*B4- (L[9]*B1+L[10]*B2+L[11]*B3+L[12]*B4);
T4:=Concatenation(T4);
T5:=B2*B1- (L[17]*B0+(L[17]- L[2] -L[6] -L[10])*B1+(L[17]- L[1] -L[5] -L[9]-1)*B2+(L[17]- L[4] -L[8] -L[12])*B3+(L[17]- L[3]-L[7]-L[11])*B4);
T5:=Concatenation(T5);
T6:=B2^2- (L[2]*B1+L[1]*B2+L[4]*B3+L[3]*B4);
T6:=Concatenation(T6);
T7:=B2*B3- (L[10]*B1+L[9]*B2+L[12]*B3+L[11]*B4);
T7:=Concatenation(T7);
T8:=B2*B4- (L[6]*B1+L[5]*B2+L[8]*B3+L[7]*B4);
T8:=Concatenation(T8);
T9:=B3*B1- (L[5]*B1+L[6]*B2+L[7]*B3+L[8]*B4);
T9:=Concatenation(T9);
T10:=B3*B2- (L[10]*B1+L[9]*B2+L[12]*B3+L[11]*B4);
T10:=Concatenation(T10);
T11:=B3^2- (L[13]*B1+L[14]*B2+L[15]*B3+L[16]*B4);
T11:=Concatenation(T11);
T12:=B3*B4- (L[18]*B0+(L[18]-L[5] -L[10] -L[13])*B1+(L[18]-L[6] -L[9] -L[14])*B2+(L[18]-L[7] -L[12] -L[15]-1)*B3+(L[18]-L[8] -L[11] -L[16])*B4);
T12:=Concatenation(T12);
T13:=B4*B1- (L[9]*B1+L[10]*B2+L[11]*B3+L[12]*B4);
T13:=Concatenation(T13);
T14:=B4*B2- (L[6]*B1+L[5]*B2+L[8]*B3+L[7]*B4);
T14:=Concatenation(T14);
T15:=B4*B3- (L[18]*B0+(L[18]-L[6] -L[9] -L[14])*B1+(L[18]-L[5] -L[10] -L[13])*B2+(L[18]-L[8] -L[11] -L[16])*B3+(L[18]-L[7] -L[12] -L[15]-1)*B4);
T15:=Concatenation(T15);
T16:=B4^2- (L[14]*B1+L[13]*B2+L[16]*B3+L[15]*B4);
T16:=Concatenation(T16);

T0:=Concatenation(T1, T2, T3, T4, T5, T6, T7, T8, T9, T10, T11, T12, T13, T14, T15, T16);

T:=AsSet(T0); 

return T; 
end; 
########################################
###########################################
EnumerationFromListRank5A2:=function(n) 
local L,k,k1,k2,k3,k4,i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12,i13, i14,i15,i16,i17,i18,i19,i20,i21,i22,i23,i24,S,T,t,V,j; 

L:=[];
S:=[];
for k1 in [1..(n-3)] do 
k2:=(n-1-2*k1)/2; 
if IsPosInt(k2) and not(k2>k1) then 

for i13 in [0..k2] do 
i11:=(1/k2)*(i13*k1); 
if IsInt(i11) then 
for i14 in [0..k2] do 
i8:=(1/k2)*(i14*k1); 
if IsInt(i8) then 
for i12 in [0..(k2-1)] do 
i7:=i12; 
for i9 in [0..k2] do 
i5:=i9; 
i10:=k2-i13-i12-i5; 
if not(i10 < 0) then 
i3:=(1/k2)*(i10*k1); 
if IsInt(i3) then 
for i6 in [0..k2] do 
i4:=(1/k2)*(i6*k1); 
if IsInt(i4) then 
for i16 in [0..k2] do 
i15:=i16+i11+i8-i7-i12-1;
if not(i15 < 0) then 
for i1 in [0..(k1-1)] do 
i2:=k1-i1-i6-i10; 
if not(i2 < 0) then 

  T:=[i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12,i13,i14,i15,i16,k1,k2]; 
  t:=CAlgRank5A2FromList(T); 
  if Size(t)=1 then 
  Add(L,T);
  fi;

fi; 
od; 
fi; 
od; 
fi; 
od; 
fi; 
fi; 
od; 
od; 
fi; 
od; 
fi; 
od; 

fi;  
od; 

#for T in S do 
#  t:=CAlgRank5A2FromList(T); 
#  if Size(t)=1 then 
#    Add(L,T); 
#  fi; 
#od; 

return L; 
end; 
##############################################
#######################################
RegularMatricesRank5A2:=function(L)
local B0,B1,B2,B3,B4, B;

B0:=IdentityMat(5);

B1:=[[ 0, 0, L[17], 0, 0], 
     [1, L[1], L[17]- L[1] -L[5] -L[9] -1, L[5], L[9]], 
     [0, L[2], L[17]- L[2] -L[6] -L[10], L[6], L[10]], 
     [0, L[3], L[17]- L[3] -L[7] -L[11], L[7], L[11]], 
     [0, L[4], L[17]- L[4]-L[8]-L[12], L[8], L[12]]];

B2:=[[ 0, L[17], 0, 0, 0], 
     [0, L[17]- L[2] -L[6] -L[10], L[2], L[10], L[6]], 
     [1, L[17]- L[1] -L[5] -L[9]-1, L[1], L[9], L[5]], 
     [0, L[17]- L[4] -L[8] -L[12], L[4], L[12], L[8]], 
     [0, L[17]- L[3]-L[7]-L[11], L[3], L[11], L[7]]];

B3:=[[ 0, 0, 0, 0, L[18]], 
     [0, L[5], L[10], L[13], L[18]-L[5] -L[10] -L[13]], 
     [0, L[6], L[9], L[14], L[18]-L[6] -L[9] -L[14]], 
     [1, L[7], L[12], L[15], L[18]-L[7] -L[12] -L[15]-1], 
     [0, L[8], L[11], L[16], L[18]-L[8] -L[11] -L[16]]];

B4:=[[ 0, 0, 0, L[18], 0], 
     [0, L[9], L[6], L[18]-L[6] -L[9] -L[14], L[14]], 
     [0, L[10], L[5], L[18]-L[5] -L[10] -L[13], L[13]], 
     [0, L[11], L[8], L[18]-L[8] -L[11] -L[16], L[16]], 
     [1, L[12], L[7], L[18]-L[7] -L[12] -L[15]-1, L[15]]];

B:=[B0,B1,B2,B3,B4];

return B;
end; 
######################################
ITAsRnk5A2Order:=function(n)
local L,l,i,V,t,j,k,S,T,B,b,G,b1,B1,g1,g2,g; 

S:=[]; 
T:=[];
L:=EnumerationFromListRank5A2(n); 
l:=Length(L);
for i in [1..l] do 
   B:=RegularMatricesRank5A2(L[i]); 
   t:=0;
   for b in B do 
    for j in [1..5] do 
     for k in [1..5] do 
       if b[j][k]<0 then 
          t:=1; 
       fi; 
     od; 
    od; 
   od;
   if t=0 then Add(S,B); fi; 
 od; 
g1:=PermutationMat((1,2),5); 
g2:=PermutationMat((1,2,3,4,5),5); 
G:=Group([g1,g2]); 
for B in S do 
 t:=0;
 for g in G do 
   B1:=[];
   for b1 in B do 
    Add(B1,b1^g); 
   od; 
   for b in T do 
   if IsEqualSet(B1,b) then   
      t:=1; 
   fi; 
   od; 
 od; 
 if t=0 then    
  Add(T,B); 
 fi;
od;

return T; 
end;  
###################################


###################################
CharTableOfTA:=function(b) 
local r,n,i,m,K,K1,k,c,q,i1,F,D,f,d,L,Q,G,a,D1,L1,l1,j,F1,f1,d1,v,d2,J,J1,L3,L4,L5,m1,m2,m3,t; 

r:=Size(b);
n:=0; 
for i in [1..r] do 
  m:=Sum(b[i][1]); 
  n:=n+m; 
od; 
K:=[];
c:=0; 
for i in [2..r] do 
 q:=Degree(MinimalPolynomial(b[i]));
 if q=r then 
   c:=1; 
   i1:=i;
   Print("polynomial",Factors(MinimalPolynomial(b[i])),"\n");
 fi;
od; 
# Print(c,"\n"); 

if c=1 then 
 F:=Factors(MinimalPolynomial(b[i1])); 
 if Length(F)=r then 
   K:=Eigenvectors(Rationals,b[i1]); 
 else 
   D:=[1]; 
   for f in F do 
    d:=Degree(f);
    if d=2 then 
     Add(D,ER(Discriminant(f)));
    fi;
   od; 
   L:=Field(D); 
   Q:=Eigenvectors(L,b[i1]);
   if Length(Q)=r then 
     K:=Q; 
   else
    f:=F[Length(F)];
    G:=TransitiveGroup(Degree(f),GaloisType(f)); 
    if not(IsAbelian(G)) then 
      K:=["Not Cyclotomic",f,G]; 
    else
      a:=PrimitiveElement(L); 
      d:=Degree(f); 
      D1:=Discriminant(f); 
#      Print("cyclotomic","\r");
      L1:=[];
      for m in [4..100] do
        if IsInt(Phi(m)/d) and IsInt(D1/m) then  
        Add(L1,Field([a,E(m)])); 
        fi;
      od; 
      L1:=AsSet(L1);
    #  Print(L1,"\n"); 
      for L in L1 do 
        l1:=Length(Factors(PolynomialRing(L),f));
    #    Print(L," ",l1,"\n");  
        if l1=d then 
          break; 
        fi; 
      od;  
    # for L in L1 do 
      Q:=Eigenvectors(L,b[i1]);
    # Print(L,Length(Q),"\r"); 
      if Length(Q)=r then 
          K:=Q; 
    #    break;
      fi; 
    # od; 
    fi; 
   fi; 
 fi; 
fi;

if c=1 and K=[] then 
  Print("cyclotomic, requires larger field","\n"); 
fi; 

if c=0 then 
 F:=[];
 for j in [2..r] do 
  F1:=AsSet(Factors(MinimalPolynomial(b[j]))); 
  F:=Concatenation(F,F1); 
  F:=AsSet(F); 
 od; 
 F:=SortedList(F); 
# Print(F,"\n");
 D:=[E(2)];
 d:=1; 
 for f in F do 
  d:=Maximum(Degree(f),d);
 od;
# Print(d,"\n"); 
 if d=1 then 
   L:=Field(D); 
 fi; 
 if d=2 then 
   for f in F do 
    if Degree(f)=2 then 
     d1:=Discriminant(f); 
     Add(D,ER(d1)); 
    fi; 
   od; 
   L:=Field(D); 
 fi;
 if d>2 then 
  G:=TransitiveGroup(Degree(f),GaloisType(f)); 
   if not(IsAbelian(G)) then 
     K:=["Not Cyclotomic",f,G];  
   fi;
 fi;
 if K=[] then   
  D:=[E(2)];  
  for f in F do 
   d:=Degree(f); 
   if d=2 then 
    d1:=Discriminant(f); 
    Add(D,ER(d1)); 
   fi; 
   if d>2 then 
    v:=Discriminant(f); 
    d1:=AbsoluteValue(v); 
    for d2 in Factors(d1) do 
    Add(D,E(d2)); 
    od; 
   fi; 
  od; 
  L:=Field(D);
#  Print(L,"\n"); 
  d1:=1;
  for f in F do 
   F1:=Factors(PolynomialRing(L),f); 
   f1:=F1[Length(F1)]; 
   d1:=Maximum(d1,Degree(f1));
  od;
#  Print(d1,"\n");     
  if d1=1 then 
   K1:=[]; 
   for k in [1..r] do 
    Q:=Eigenvectors(L,b[k]); 
    #Print(k,Q,"\n"); 
    d1:=DiagonalOfMat(Q*b[k]*Q^(-1)); 
    Add(K1,d1);  
   od;
   #Print(K1,"\n");
   J:=DiagonalOfMat(IdentityMat(r));
   #Print(J*K1,"\n"); 
   if not(J*K1=[n,0,0,0,0]) then 
     G:=SymmetricGroup([2..r]); 
     L3:=Orbit(G,K1[3],Permuted); 
     L4:=Orbit(G,K1[4],Permuted); 
     L5:=Orbit(G,K1[5],Permuted); 
     t:=0;
     for m1 in L3 do 
     for m2 in L4 do 
     for m3 in L5 do 
     Q:=[K1[1],K1[2],m1,m2,m3];
     J1:=J*Q; 
     if J1=[n,0,0,0,0] then
      K:=TransposedMat(Q); 
      t:=1;
      break;
     fi;
     od;
     if t=1 then 
      break;
     fi;
     od;
     if t=1 then 
      break; 
     fi;
     od; 
   fi;
  fi; 
 fi;
fi; 

return K; 
end;

#####################################
DegreesOfTA:=function(b)
local r,D,i;

r:=Length(b); 
D:=[];
for i in [1..r] do 
  D[i]:=b[i][1][i];
od; 

return D;
end;
#####################################
# Another name for character table of 
# a commutative table algebra is first 
# eigenmatrix. 
#####################################
FirstEigenmatrixOfTA:=function(b)
local Q;

Q:=CharTableOfTA(b); 

return Q;
end;

#####################################
# Note that the second eigenmatrix is 
# computed from the first eigenmatrix.
#####################################
SecondEigenmatrixOfTA:=function(b)
local Q,n,P; 

P:=FirstEigenmatrixOfTA(b);
n:=Sum(P[1]); 
Q:=n*P^(-1);

return Q; 
end; 
#####################################
MultiplicitiesOfTA:=function(b)
local n,P,M,Q;

P:=FirstEigenmatrixOfTA(b);
n:=Sum(P[1]); 
Q:=n*P^(-1); 
M:=Q[1]; 

return M;
end;
#####################################
ClosedSubsetsOfTA:=function(b)
local r,C0,C1,c,C2,C,i,j,k,a,b1,D;

r:=Length(b);
a:=Algebra(Rationals,b);
b1:=Basis(a,b); 
C:=[];
C0:=Combinations([2..r]); 
C1:=[];
for c in C0 do 
  Add(C1,Concatenation([1],c));
od; 
for c in C1 do 
D:=[];
for j in c do 
for k in c do 
  C2:=Coefficients(b1,b1[j]*b1[k]); 
  for i in [1..r] do 
  if not(C2[i]=0) then 
    AddSet(D,i); 
  fi;
  od;
od;
od;

if D=c then
  Add(C,D);
fi; 
od;  

return C;
end;
################################### 
FusionsOfTA:=function(b)
local S,P,D,p,p1,b1,p2,d,a,r;

S:=[];
r:=Length(b);
P:=PartitionsSet([1..r]);
for p in P do 
 if not(p=[]) then 
 D:=[]; 
 for p1 in p do
 b1:=0*b[1]; 
  for p2 in p1 do 
   b1:=b1+b[p2]; 
  od; 
  Add(D,b1); 
 od; 
 a:=Algebra(Rationals,D);
 d:=Dimension(a); 
 if d=Length(p) then 
  Add(S,p); 
 fi; 
 fi;
od;

return S;
end;