Guide to contributing: Code

The code in this section computes the Cartan matrix, the decomposition matrix and the Loewy structure of the projective indecomposable modules. The output is formatted in LaTeX, so it can be copied and pasted easily on the relevant block page.

Magma code

The following code, written by Claudio Marchi and Cesare G. Ardito, can usually be run on the free [Magma online calculator], as long as the involved group is small.

LoewyStructure:= function(W,p,k,S,irrs)
  Z:=W;
  t:=0;
  maxlength:=100;
  M:=ZeroMatrix(Integers(),maxlength,maxlength);
  repeat                
    t:=t+1;
    J:=JacobsonRadical(W);
    V:=W/J;
    V:=ChangeRing(V,GF(p^k));
    if IsDecomposable(V) then
      indsum:=IndecomposableSummands(V);
      for i in [1..#indsum] do
        for j in S do
          V7:=indsum[i];
          U:=irrs[j];
          U:=ChangeRing(U,GF(p^k));
          V:=ChangeRing(V7,GF(p^k));
          if IsIsomorphic(V,U) then
            M[t][i]:=j;                                    
            continue i;
          end if;
        end for;
      end for;    
    else
      for j in S do
        U:=irrs[j];
        U:=ChangeRing(U,GF(p^k));
        V:=ChangeRing(V,GF(p^k));
        if IsIsomorphic(V,U) then
          M[t][1]:=j;
          break j;
        end if;
      end for;
    end if;    
    W:=J;
    if IsIrreducible(W) then
      t:=t+1;
      for j in S do
        U:=irrs[j];
        U:=ChangeRing(U,GF(p^k));
        V:=ChangeRing(W,GF(p^k));
        if IsIsomorphic(V,U) then
          M[t][1]:=j;
          break j;
        end if;
      end for;    
    end if;
  until IsIrreducible(W);
  printf("\n");
  for i in [1..maxlength] do
    flag:=0;
    for j in [1..maxlength] do
      if(not (M[i][j] eq 0)) then 
        flag:=1;
        printf "S_\{%o\} ", M[i][j]; 
      end if;
    end for;
    if(flag eq 1) then
      printf("\\\\ \n");
    end if;
  end for;
  print "The structure above is the Loewy structure of";
  return Z;
  end function;

PrintLatex:= function(M)
  for i in [1..NumberOfRows(M)-1] do
    for j in [1..NumberOfColumns(M)-1] do
      printf "%o & ", M[i,j];
    end for;
    printf "%o \\\\ \n", M[i,NumberOfColumns(M)];
  end for;
  for j in [1..NumberOfColumns(M)-1] do
    printf "%o & ", M[NumberOfRows(M),j];
  end for;
printf "%o \n", M[NumberOfRows(M),NumberOfColumns(M)];
return " ";
end function;

/* Input data: a group, a prime and an exponent such that GF(p^n) contains all the necessary roots of unity */
G:=SmallGroup(2,1); 
p:=2;
splittingexponent:=1;
/* End input data*/

PrintLatex(AbsoluteCartanMatrix(G,GF(p^splittingexponent)));
printf("\n");
PrintLatex(DecompositionMatrix(G,GF(p^splittingexponent)));
irrs:=AbsolutelyIrreducibleModules(G, GF(p^splittingexponent));
P:=ProjectiveIndecomposableModules(G, GF(p^splittingexponent));
for i in [1..#P] do
  LoewyStructure(P[i],p,splittingexponent,[1..#irrs],irrs);
end for;

Note that, if [math]G[/math] has more than one block, some manual work on the output is needed to isolate the relevant submatrices and indecomposable projective modules that belong to the block that is being investigated.