int_Matrix PreRootDatum::Cartan_matrix() const
{
int_Matrix Cartan(d_roots.size(),d_coroots.size());
for (weyl::Generator i = 0; i < d_roots.size(); ++i)
for (weyl::Generator j = 0; j < d_coroots.size(); ++j)
Cartan(i,j) = d_roots[i].dot(d_coroots[j]);
return Cartan;
}
matrix* Resmat(matrix* roots)
{ lie_Index i,j,k,r=Lierank(grp),s=Ssrank(grp), n=roots->nrows;
vector* root_norms=Simproot_norms(grp);
entry* norms=root_norms->compon;
/* needed to compute $\<\lambda,\alpha[i]>$ */
matrix* root_images=Matmult(roots,Cartan()),* result=mkmatrix(r,r);
entry** alpha=roots->elm,** img=root_images->elm,** res=result->elm;
for (i=0; i<r; i++) for (j=0; j<r; j++) res[i][j]= i==j;
/* initialise |res| to identity */
for (j=0; j<n; j++) /* traverse the given roots */
{ entry* v=img[j], norm=(checkroot(alpha[j]),Norm(alpha[j]));
for (k=s-1; v[k]==0; k--) {}
if (k<j)
error("Given set of roots is not independent; apply closure first.\n");
if (v[k]<0)
{ for (i=j; i<n; i++) img[i][k]= -img[i][k];
for (i=k-j; i<s; i++) res[i][k]= -res[i][k];
}
while(--k>=j)
/* clear |v[k+1]| by unimodular column operations with column~|j| */
{
entry u[3][2]; lie_Index l=0;
u[0][1]=u[1][0]=1; u[0][0]=u[1][1]=0;
u[2][1]=v[k]; u[2][0]=v[k+1];
if (v[k]<0) u[2][1]= -v[k], u[0][1]= -1; /* make |u[2][1]| non-negative */
do /* subtract column |l| some times into column |1-l| */
{ entry q=u[2][1-l]/u[2][l]; for (i=0; i<3; i++) u[i][1-l]-=q*u[i][l];
} while (u[2][l=1-l]!=0);
if (l==0) for (i=0; i<2; i++) swap(&u[i][0],&u[i][1]);
{ for (i=j; i<n; i++) /* combine columns |k| and |k+1| */
{ entry img_i_k=img[i][k];
img[i][k] =img_i_k*u[0][0]+img[i][k+1]*u[1][0];
img[i][k+1]=img_i_k*u[0][1]+img[i][k+1]*u[1][1];
}
for (i=k-j; i<s; i++)
{ entry res_i_k=res[i][k];
res[i][k]=res_i_k*u[0][0]+res[i][k+1]*u[1][0];
res[i][k+1]=res_i_k*u[0][1]+res[i][k+1]*u[1][1];
}
}
}
for (i=0; i<s; i++)
{ lie_Index inpr= norms[i]*alpha[j][i]; /* this is $(\omega_i,\alpha[j])$ */
if (inpr%norm!=0) error("Supposed root has non-integer Cartan product.\n");
res[i][j]=inpr/norm; /* this is $\<\omega_i,\alpha[j]>$ */
}
}
freemem(root_norms); freemem(root_images); return result;
}
matrix* Weyl_root_orbit(entry* v)
{ lie_Index i,j,r=Lierank(grp),s=Ssrank(grp);
entry* x=mkintarray(r); matrix* orbit, *result; entry** m;
lie_Index dc=Detcartan();
mulvecmatelm(v,Cartan()->elm,x,s,r);
orbit=Weyl_orbit(x,NULL);
result=mkmatrix(orbit->nrows,s); m=result->elm;
mulmatmatelm(orbit->elm,Icartan()->elm,m,orbit->nrows,s,s);
freemem(orbit);
for (i=0; i<result->nrows; ++i)
for (j=0; j<s; ++j) m[i][j]/=dc;
return result;
}
