algsplit

setrand(1)

vec2mat(d,n,V) =
{
  my(M = matrix(d,d), x = 'x, c);
  for(i=0,d^2*n-1,
    c = V[i+1];
    M[i\(n*d) + 1, (i\n)%d + 1] += c*x^(i%n)
  );
  c = polcoeff(M[1,1],0);
  for(i=2,d,M[i,i]+=c);
  M
};

mat2vec(d,n,M) =
{
  my(V = vector(d^2*n), c = polcoeff(M[1,1],0));
  for(i=2,d,M[i,i]-=c);
  for(i=1,d,
    for(j=1,d,
      for(k=0,n-1,
        V[n*d*(i-1) + n*(j-1) + k + 1] = polcoeff(M[i,j],k);
      )
    )
  );
  V
};

matalg(d,n,p) =
{
  my(pol = ffinit(p,n), mtx, N=d^2*n, x, basis, mt=vector(N));
  basis = vector(N,i,vec2mat(d,n,vector(N,k,k==i)));
  basis = basis*Mod(1,p)*Mod(1,pol);
  for(i=1,N,
    x = basis[i];
    mtx = matconcat(vector(N,j,mat2vec(d,n,liftall(x*basis[j]))~));
    mt[i] = mtx;
  );
  algtableinit(mt,p)
};

smallchg(mt,p) =
{
  my(i,j,N=#mt,c);
  if(N==1, return(mt));
  if(p, c=random([1,p-1]), c=1);
  i = random([2,N]);
  j=i;
  while(j==i,j=random([1,N]));
  for(k=1,N,
    mt[k][,i] += c*mt[k][,j];
    mt[k][j,] -= c*mt[k][i,];
  );
  mt[i] += c*mt[j];
  if(p,mt%p,mt)
};

chg(mt,nb=3*#mt,p=0) =
{
  for(c=1,nb,mt=smallchg(mt,p));
  mt
};

test2(d,n,p,nb=1) =
{
  my(al,mt,mt2,res,map,x,y,Mx,My);
  setrand(1);
  al = matalg(d,n,p);
  mt = algmultable(al);
  mt2 = mt;
  for(c=1,nb,
    if(c>1, mt2 = chg(mt2,#mt\4 + 5,p));
    al = algtableinit(mt2,p);
    res = algsplit(al,'t);
    map = res[1];
    mapi = res[2];
    x = algrandom(al,10);
    y = algrandom(al,10);
    Mx = map*x;
    My = map*y;
    if(Mx*My != map*algmul(al,x,y),  print("FAIL"); return(0));
  );
  [al,map,mapi]
};

print("M_2(F_4)");
test2(2,2,2,200);

print("M_2(F_9)");
test2(2,2,3,100);

q = 31;
print("M_2(F_31^2)");
test2(2,2,q,30);

p = nextprime(10^20);
print("M_2(F_p^2)");
test2(2,2,p,30);

print("M_d(F_31^n)");
for(d=1,4,for(n=1,min(18\(d^2),10),print("d=",d," n=",n);if(!test2(d,n,q,10), break(2))));

isblock(M,L) =
{
  my(d,mini,nxt);
  d = #M[,1];
  mini=L[1];
  nxt=2;
  for(j=1,d,
    if(j>mini,
      mini += L[nxt];
      nxt += 1;
    );
    for(i=mini+1,d,
      if(M[i,j]!=0,return(0));
    );
  );
  1
};
matblock(L,n,p) =
{
  d = sum(i=1,#L,L[i]);
  al0 = matalg(d,n,p);
  basis = [b | b<-Vec(matid(d^2*n)), isblock(vec2mat(d,n,b),L)];
  basis = Mat(basis);
  algsubalg(al0,basis)[1];
};

print("examples from documentation");
al0 = alginit(nfinit(y^2+7), [-1,-1]);
al = algtableinit(algmultable(al0), 3); \\isomorphic to M_2(F_9)
[map,mapi] = algsplit(al, 'a);
x = [1,2,1,0,0,0,0,0]~; fx = map*x
y = [0,0,0,0,1,0,0,1]~; fy = map*y
map*algmul(al,x,y) == fx*fy
map*mapi[,6]

print("bad input");
algsplit("toto");
algsplit(alginit(nfinit('y),[-1,-1]));
algsplit(algtableinit([matid(3),[0,0,0;1,1,0;0,0,1],[0,0,0;0,0,0;1,0,0]],2));
algsplit(algtableinit([matid(2),[0,0;1,1]],2));
algsplit(matblock([2,1,1,1],1,q));
algsplit(matblock([1,2,3],1,q));
{mt=[[1, 0, 0, 0, 0, 0, 0, 0, 0; 0, 1, 0, 0, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0,
0, 0; 0, 0, 0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0,
0, 0; 0, 0, 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0,
0, 1], [0, 0, 0, 1, 0, 0, 0, 0, 0; 1, 0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 1,
0, 0, 0; 0, 1, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0, 1; 0, 0, 0, 1, 0, 0,
1, 0, 0; 0, 1, 1, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0, 0, 0; 0, 1, 0, 0, 0, 0,
0, 1, 0], [0, 0, 0, 0, 0, 0, 1, 0, 1; 0, 0, 0, 0, 1, 1, 0, 0, 0; 1, 0, 0, 0, 1,
0, 0, 0, 0; 0, 1, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 1, 0,
0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 1, 0, 0,
0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 1, 0, 1; 0, 0, 0, 0,
0, 0, 0, 0, 1; 1, 0, 0, 0, 0, 0, 0, 0, 0; 0, 1, 1, 0, 0, 0, 0, 0, 0; 0, 0, 1, 0,
0, 0, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0, 1; 0, 0, 0, 0,
0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 1, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0, 1, 0; 0, 1, 0,
0, 0, 0, 0, 1, 0; 0, 0, 0, 1, 0, 0, 0, 0, 0; 1, 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0,
0, 0, 1, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0, 1; 0, 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0,
1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 1, 0, 0, 0; 0, 1, 0, 0, 0, 0, 0, 0, 0; 0, 1,
0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 1, 0, 1; 0, 0, 0, 0, 1, 0, 0, 0, 0; 1, 0,
0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0, 1; 0, 1, 1, 0, 0, 0, 0, 0, 0; 0, 0,
0, 0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0, 1; 0,
0, 0, 1, 0, 0, 1, 0, 0; 0, 0, 0, 0, 1, 1, 0, 0, 0; 0, 1, 1, 0, 0, 0, 0, 0, 0; 0,
1, 0, 0, 0, 0, 0, 1, 0; 1, 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0, 0; 0,
0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 0, 1, 1, 0, 0, 0;
0, 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 1, 0, 0;
0, 0, 0, 0, 0, 0, 0, 0, 1; 0, 1, 1, 0, 0, 0, 0, 0, 0; 1, 0, 0, 0, 0, 1, 0, 0, 0;
0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 1, 0; 0, 0, 0, 1, 0, 0, 0, 0,
0; 0, 0, 0, 0, 0, 0, 0, 0, 1; 0, 0, 0, 0, 1, 1, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 1,
0; 0, 1, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0,
0; 1, 0, 0, 0, 1, 0, 0, 0, 0]]};
algsplit(algtableinit(mt,2));