matrix* Weyl_orbit(entry* v, matrix** orbit_graph)
{ lie_Index i,j,k,r=Lierank(grp),s=Ssrank(grp);
matrix* result; entry** m;
lie_Index level_start=0, level_end=1, cur=1;
{ entry* lambda=mkintarray(r);
copyrow(v,lambda,r); make_dominant(lambda);
result=mkmatrix(bigint2entry(Orbitsize(lambda)),r);
copyrow(lambda,result->elm[0],r); freearr(lambda);
if (orbit_graph!=NULL) *orbit_graph=mkmatrix(result->nrows,s);
}
m=result->elm;
while (level_start<level_end)
{
for (k=level_start; k<level_end; ++k)
for (i=0; i<s; ++i)
if (m[k][i]>0) /* only strictly cross walls, and from dominant side */
{ w_refl(m[k],i);
for (j=level_end; j<cur; ++j)
if (eqrow(m[k],m[j],s)) break;
if (orbit_graph!=NULL)
{ (*orbit_graph)->elm[k][i]=j; (*orbit_graph)->elm[j][i]=k; }
if (j==cur)
{ assert(cur<result->nrows);
copyrow(m[k],m[cur++],r);
}
w_refl(m[k],i);
}
else if (m[k][i]==0 && orbit_graph!=NULL) (*orbit_graph)->elm[k][i]=k;
level_start=level_end; level_end=cur;
}
return result;
}
static lie_Index n_parts(lie_Index n)
{ lie_Index i,k,np; entry* c=mkintarray(n+1); /* coefficients */
if (n>121) error("Too many partitions to generate.\n");
for (i=0; i<=n; ++i) c[i]=1; /* initialise to ${1\over1-X}$ */
for (i=2; i<=n; ++i)
for (k=i; k<=n; ++k) c[k]+=c[k-i]; /* multiply by ${1\over1-X^i}$ */
np=c[n]; freearr(c); return np;
}
poly* Vdecomp(poly* p)
{ lie_Index i,r=Lierank(grp);
poly* result=poly_null(r);
cur_expon=mkintarray(r); /* large enough */
for (i=0; i<p->nrows; ++i)
result=Addmul_pol_pol_bin(result,vdecomp_irr(p->elm[i]),p->coef[i]);
freearr(cur_expon);
return result;
}
bigint* Orbitsize(entry* w)
{ lie_Index i,d,s=Ssrank(grp); entry* x=mkintarray(s),* y=x; bigint* result=one;
copyrow(w,x,s); make_dominant(x);
if (type_of(grp)==SIMPGRP) return simp_worbitsize(x,&grp->s);
for (i=0; i<grp->g.ncomp; ++i,y+=d)
{ simpgrp* g=Liecomp(grp,i); d=g->lierank;
result=mult(result,simp_worbitsize(y,g));
}
freearr(x); return result;
}
poly* LR_tensor_irr(entry* lambda,entry * mu, _index n)
{ _index i,j;
entry* nu; entry** T;
if (n==0) return poly_one(0);
{ nu=&mkintarray(n+1)[1]; copyrow(lambda,nu,n); nu[-1]=lambda[0]+mu[0];
T=alloc_array(entry*,n+1);
for (i=0;i<=n;++i) /* allocate row |T[i]| and place sentinel before it */
{ T[i]= &mkintarray(mu[i==0?0:i-1]+1)[1]; T[i][-1]=n-1-i; }
for (i=0,j=mu[0]-1; j>=0; --j)
{ while (i<n && mu[i]>j) ++i; /* find first |i| with |mu[i]<=j| */
T[i][j]=-1; /* place sentinel at bottom of column |j| */
}
}
wt_init(n); /* prepare to collect terms with exponents of size~|n| */
{ j=-1; for (i=n-1; i>0 && mu[i]==0; --i) {} /* move to initial position */
recurse: /* recursive starting point; */
if (++j>=mu[i] &&(j=0,--i<0)) /* move to next empty position, if any */
wt_ins(nu,one,false); /* if not, |T| is full; contribute |nu| once */
else
{ _index k= T[i+1][j]; entry prev= nu[k];
do
{ while (nu[++k]==prev) {} /* find next |k| with |nu[k]<nu[@t$k'$@>]| */
++nu[T[i][j]=k]; goto recurse;
/* insert |k| into |T| and extend partition |nu|; recurse */
resume: prev= --nu[k=T[i][j]];
/* restore |k| and |nu|; set |prev=nu[k]| */
} while (prev>nu[T[i][j-1]]);
/* if so, there are still corners of |nu| to try */
}
if (j==0) j= ++i<n?mu[i]:0; /* return to end of row below if necessary */
if (--j>=0) goto resume; /* do return jump unless empty row is reached */
}
{ --nu; freearr(nu);
for (i=0;i<=n;i++) { entry* t=&T[i][-1]; freearr(t); }
freearr(T);
}
return wt_collect(); /* return sum of all contributed terms */
}
entry Schur_char_val(entry* lambda, entry* mu, lie_Index l, lie_Index m)
{ lie_Index i; entry sum=0;
while (l>0 && lambda[l-1]==0) --l; /* get reduced form of~|lambda| */
if (l<=1) return 1; /* trivial character */
if (l>lambda[0]) /* then better work with the transpose partition */
{ vector* tr=Trans_part(lambda,l);
entry ch=Schur_char_val(tr->compon,mu,lambda[0],m); freemem(tr);
return Sign_part(mu,m)*ch;
}
{ entry* lambda_prime=mkintarray(4*l)
,* sigma=lambda_prime+l,* pos=sigma+l,* nu=pos+l;
/* 4 length-|l| arrays */
boolean sg=true; /* positive sign */
copyrow(lambda,lambda_prime,l);
/* |lambda| might be alias of |mu|, but |lambda_prime| is not */
for (i=0; i<l; ++i) pos[i]=sigma[i]=i;
/* |sigma| is the permutation; |pos| records its swap sequence */
do
{ copyrow(lambda_prime,nu,l);
{ lie_Index i; for (i=1; i<l; ++i) if (nu[i]>nu[i-1]) /* skip most cases */
{ entry nui=nu[i]; lie_Index j=i;
do nu[j]=nu[j-1]; while (--j>0 && nui>nu[j-1]); nu[j]=nui;
}
}
sum+= sg ? Young_char_val(nu,mu,l,m) : -Young_char_val(nu,mu,l,m);
{ lie_Index i=0,j;
do
{
{ lambda_prime[i]-=sigma[i];
if ((j=pos[i])<i) { swap(&sigma[i],&sigma[j]); sg=!sg;}
}
do
if(--j<0) break; /* tried all positions for this |i| */
while (lambda_prime[i]+sigma[j]<0);
} while (j<0 && ++i<l);
if (i==l) break;
do /* now |j>=0| and |sigma[j]| can move validly to |sigma[i]| */
{
{ if ((pos[i]=j)<i) { swap(&sigma[i],&sigma[j]); sg=!sg;}
lambda_prime[i]+=sigma[i]; /* this becomes non-negative */
}
if (--i<0) break;
for (j=i; lambda_prime[i]+sigma[j]<0; --j) {} /* this leaves |j>=0| */
} while (true);
}
} while (true);
freearr(lambda_prime);
}
return sum;
}
bigint* MN_char_val(entry* lambda, entry* mu, lie_Index l, lie_Index m)
{ bigint* value=null; lie_Index n=check_part(lambda,l),m2;
if (n==0) return one;
while (lambda[l-1]==0) --l; while (mu[m-1]==0) --m;
for (m2=m; m2>0 && mu[m2-1]==1; --m2) {} /* number of parts $\mu_i\geq2$ */
{ entry* save=mkintarray(2*n),* lambda_prime=save+n;
int i, j, d=lambda[0]+l, k=0; /* sum of leg lengths */
boolean* edge=alloc_array(boolean,2*d);
enum {hor, vert};
{ int r=l-1,c=0; /* current column number */
for (j=0; r>=0; --r)
{ while (c<lambda[r]) { edge[j++]=hor; ++c; } /* columns of length |r| */
edge[j++]=vert; /* row |r|, of length |c==lambda[r]| */
}
}
for (i=0; i<m2; ++i)
{ for (j=0; j+mu[i]<d; ++j)
if (edge[j]==hor && edge[j+mu[i]]==vert) break;
if (j+mu[i]==d) return null; /* no hook of size |mu[i]| was found */
}
{ i=0; /* index into |mu| */
recurse:
if (i<m2)
{ int r=mu[i];
for (j=1; j<r; ++j) k+=edge[j]; /* leg length of hook first tried */
for (j=0; j+r<d; ++j)
{ if (edge[j]==hor && edge[j+r]==vert)
{ edge[j]=vert; edge[j+r]=hor; save[i++]=j; goto recurse;
resume: j=save[--i]; r=mu[i]; edge[j]=hor; edge[j+r]=vert;
}
k+= edge[j+r]-edge[j+1]; /* adjust |k| for hook tried next */
}
while (++j<d) k-= edge[j]; /* restore |k| */
}
else
{ int r=l,c=0,s=0; /* size of |lambda_prime| */
for (j=0; r>0; )
if (edge[j++]==vert) s+=lambda_prime[--r]=c; else ++c;
/* build |lambda_prime| from edges */
value= k%2==0 ? add(value,n_tableaux(lambda_prime,l))
: sub(value,n_tableaux(lambda_prime,l)) ;
}
if (i>0) goto resume;
}
freearr(edge); freearr(save);
}
return value;
}
matrix* Partitions(lie_Index n)
{ matrix* result=mkmatrix(n_parts(n),n);
if (n>0)
{ entry* lambda=mkintarray(n),** res=result->elm; lie_Index i=0,j;
lambda[0]=n;
for(j=1;j<n;j++) lambda[j]=0; /* initialise |lambda| to $[n,0,0,\ldots]$ */
do copyrow(lambda,res[i++],n); while(Nextpart(lambda,n));
freearr(lambda);
}
return result;
}
matrix* Permutations(entry* v,lie_Index n)
{ lie_Index N=1; entry* w=mkintarray(n); copyrow(v,w,n); sortrow(w,n);
{ lie_Index i=0,j=n-1; while (i<j) swap(&w[i++],&w[j--]); }
/* increasing order */
{ lie_Index i=0, mult=1;
while (++i<n) { N*=i+1; if (w[i]>w[i-1]) mult=1; else N /= ++mult; }
}
{ matrix* result=mkmatrix(N,n); lie_Index i=0;
do copyrow(w,result->elm[i++],n); while (Nextperm(w,n));
freearr(w); return result;
}
}
matrix* Tableaux(entry* lambda, lie_Index l)
{ bigint* nt=n_tableaux(lambda,l); lie_Index n=check_part(lambda,l);
matrix* result=mkmatrix(bigint2entry(nt),n);
entry** res=result->elm,* t=mkintarray(n);
freemem(nt);
{ lie_Index i=0,j,k;
for (j=1; j<=l; ++j) for (k=lambda[j-1]; k>0; --k) t[i++]=j;
}
{ lie_Index i=0; do copyrow(t,res[i++],n); while(Nexttableau(t,n)); }
freearr(t); 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;
}
bigint* n_tableaux(entry* lambda, lie_Index l)
{ lie_Index i,j,k=0; entry* h; bigint* res=copybigint(one,NULL);
do if (--l<=0) return one;
while (lambda[l]==0); /* find last non-zero part */
h=mkintarray(lambda[0]);
for(j=0; j<lambda[0]; ++j) h[j]=0; /* accumulated column heigths */
for(i=l; i>=0; --i)
{ entry li=lambda[i]-1;
for(j=0; j<=li; ++j) res=mul1(res,++k); /* part of factorial */
for(j=0; j<=li; ++j) div1(res,(++h[j])+li-j); /* divide by hook lengths */
}
freearr(h); return res;
}
local void long_close(matrix* m, lie_Index first, lie_Index last)
{ lie_Index i,j; entry* root_i,* root_j,* t=mkintarray(s);
for (i=first; i<last; ++i)
{ root_i=m->elm[i]; if (Norm(root_i)>1) continue;
for (j=i+1; j<last; ++j)
{ root_j=m->elm[j]; if (Norm(root_j)>1) continue;
subrow(root_i,root_j,t,s);
if (isroot(t))
if (isposroot(t))
{ copyrow(t,root_i,s); break; } /* need not consider more |j|'s */
else add_xrow_to(root_j,-1,root_i,s);
}
}
freearr(t);
}
local void fundam(matrix* roots, lie_Index first, lie_Index* last)
{ lie_Index i,j,d; boolean changed;
entry* t=mkintarray(s); matrix mm,* m=&mm;
mm.elm=&roots->elm[first]; mm.nrows=*last-first; mm.ncols=roots->ncols;
for (i=m->nrows-1; i>0; changed ? Unique(m,cmpfn),i=m->nrows-1 : --i)
{ entry* root_i=m->elm[i]; changed=false;
for (j=i-1; j>=0; j--)
{ entry* root_j=m->elm[j]; entry c=Cart_inprod(root_j,root_i);
if (c==2 && eqrow(root_j,root_i,s))
{ cycle_block(m,j,m->nrows--,1); root_i=m->elm[--i];
}
else if (c>0)
{ changed=true;
{ copyrow(root_j,t,s); add_xrow_to(t,-c,root_i,s);
if (isposroot(t)) copyrow(t,root_j,s);
else
{ j=i; c=Cart_inprod(root_i,root_j);
copyrow(root_i,t,s); add_xrow_to(t,-c,root_j,s);
if (isposroot(t)) copyrow(t,root_i,s);
else
{ lie_Index k; entry* ln,* sh; /* the longer and the shorter root */
if (Norm(root_i)>Norm(root_j))
ln=root_i, sh=root_j; else ln=root_j, sh=root_i;
switch (Norm(ln))
{ case 2: subrow(ln,sh,sh,s); /* |sh=ln-sh| */
add_xrow_to(ln,-2,sh,s); /* |ln=ln-2*sh| */
break; case 3: /* |grp=@t$G_2$@>| now */
for (k=0; sh[k]==0; ++k) {} /* find the place of this $G_2$ component */
sh[k]=1; sh[k+1]=0; ln[k]=0; ln[k+1]=1;
/* return standard basis of full system */
break; default: error("problem with norm 1 roots\n");
}
}
}
}
}
}
}
cycle_block(roots,first+mm.nrows,roots->nrows,d=*last-first-mm.nrows);
*last-=d; roots->nrows-=d;
freearr(t);
}
poly* alt_Wsum(poly* p)
{ lie_Index i,k=0,r=p->ncols; poly* result; entry** res,*rho=mkintarray(r);
p=Alt_dom(p); for (i=0; i<r; ++i) rho[i]=1;
for (i=0; i<p->nrows; ++i) add_xrow_to(p->elm[i],1,rho,r);
result=mkpoly(p->nrows*bigint2entry(Worder(grp)),r); res=result->elm;
for (i=0; i<p->nrows; ++i)
{ lie_Index j,l; matrix* orbit=Weyl_orbit(p->elm[i],NULL); entry** x=orbit->elm;
bigint* c=p->coef[i],* min_c=sub(null,c);
for (j=0; j<orbit->nrows; ++j)
{ subrow(*x,rho,res[k],r); l=make_dominant(*x++)%2;
result->coef[k]= l ? min_c : c; setshared(result->coef[k]); ++k;
}
freemem(orbit);
}
freearr(rho);
assert(k==result->nrows);
return result; /* not sorted, but rows are unique */
}
matrix* simp_proots(simpgrp* g)
{ if (g->roots!=NULL) return g->roots;
{ _index r=g->lierank,l,i,last_root;
entry** cartan=simp_Cartan(g)->elm;
entry** posr=(g->roots=mkmatrix(simp_numproots(g),r))->elm;
entry* level=(g->level=mkvector(simp_exponents(g)[r-1]+1))->compon;
entry* norm=(g->root_norm=mkvector(g->roots->nrows))->compon;
entry* alpha_wt=mkintarray(r);
/* space to convert roots to weight coordinates */
setlonglife(g->roots),
setlonglife(g->level),
setlonglife(g->root_norm); /* permanent data */
{ _index i,j; for (i=0; i<r; ++i) for (j=0; j<r; ++j) posr[i][j] = i==j;
level[0]=0; last_root=r;
for (i=0; i<r; ++i) norm[i]=1; /* norms are mostly |1| */
switch (g->lietype) /* here are the exceptions */
{ case 'B':
for (i=0; i<r-1; ++i) norm[i]=2; /* $2,2,\ldots,2,1$ */
break; case 'C': norm[r-1]=2; /* $1,1,\ldots,1,2$ */
break; case 'F': norm[0]=norm[1]=2; /* $2,2,1,1$ */
break; case 'G': norm[1]=3; /* $ 1,3$ */
}
}
for (l=0; last_root>level[l]; ++l)
{ level[l+1]=last_root; /* set beginning of a new level */
for (i=level[l]; i<level[l+1]; ++i)
{ _index j,k; entry* alpha=posr[i]; mulvecmatelm(alpha,cartan,alpha_wt,r,r);
/* get values $\<\alpha,\alpha_j>$ */
for (j=0; j<r; ++j) /* try all fundamental roots */
{ entry new_norm;
{ if (alpha_wt[j]<0) /* then $\alpha+\alpha_j$ is a root; find its norm */
if (norm[j]==norm[i]) new_norm=norm[j]; /* |alpha_wt[j]==-1| */
else new_norm=1; /* regardless of |alpha_wt[j]| */
else if (norm[i]>1 || norm[j]>1) continue; /* both roots must be short now */
else if (strchr("ADE",g->lietype)!=NULL) continue;
/* but long roots must exist */
else if (alpha_wt[j]>0)
if (g->lietype!='G'||alpha_wt[j]!=1) continue; else new_norm=3;
/* $[2,1]\to[3,1]$ for $G_2$ */
else if (alpha[j]==0) continue;
/* $\alpha-\alpha_j$ should not have a negative entry */
else
{
{ --alpha[j];
for (k=level[l-1]; k<level[l]; ++k)
if (eqrow(posr[k],alpha,r)) break;
++alpha[j];
if (k==level[l]) continue;
}
new_norm=2; }
}
++alpha[j]; /* temporarily set $\alpha\K\alpha+\alpha_j$ */
for (k=level[l+1]; k<last_root; ++k)
if (eqrow(posr[k],alpha,r)) break;
/* if already present, don't add it */
if (k==last_root)
{ norm[last_root]=new_norm; copyrow(alpha,posr[last_root++],r); }
--alpha[j]; /* restore |alpha| */
}
}
}
freearr(alpha_wt); return g->roots;
}
}
poly* MN_char(entry* lambda, lie_Index l)
{ lie_Index n=check_part(lambda,l);
if (n==0) return poly_one(0); /* the character of $\Sym0$ */
while (lambda[l-1]==0) --l; /* minimise |l| */
wt_init(n); /* get ready for accumulating contributions to the character */
{
entry* mu=mkintarray(3*n),* save=mu+n,* lambda_prime=save+n;
int i, j, r, d=lambda[0]+l, k=0; /* sum of leg lengths */
boolean* edge=alloc_array(boolean,2*d-2),* candidate=edge+d-2;
/* lie_Index |2<=r<d| */
enum {hor, vert}; /* values used for |edge| */
for (i=0; i<n; ++i) mu[i]=0;
{ int r=l-1,c=0; /* current column number */
for (j=0; r>=0; --r)
{ while (c<lambda[r]) { edge[j++]=hor; ++c; } /* columns of length |r| */
edge[j++]=vert; /* row |r|, of length |c==lambda[r]| */
}
}
for (r=2; r<d; ++r)
{ for (j=0; j+r<d; ++j)
if (edge[j]==hor && edge[j+r]==vert) break;
candidate[r]= j+r<d;
}
{ i=0; /* index of last entry that was set in~|mu| */
for (r=d-1; r>1; --r) /* try hooks of size |r| */
if (candidate[r])
{ recurse: /* recursive starting point */
{ for (j=1; j<r; ++j) k+=edge[j]; /* leg length of hook first tried */
for (j=0; j<d-r; ++j)
{ if (edge[j]==hor && edge[j+r]==vert)
{ edge[j]=vert; edge[j+r]=hor; mu[i]=r; save[i++]=j; goto recurse;
resume: j=save[--i]; r=mu[i]; mu[i]=0; edge[j]=hor; edge[j+r]=vert;
}
k+= edge[j+r]-edge[j+1]; /* adjust |k| for hook tried next */
}
while (++j<d) k-= edge[j]; /* restore |k| */
}
}
}
{ int r=l,c=0,s=0; /* size of |lambda_prime| */
for (j=0; r>0; )
if (edge[j++]==vert) s+=lambda_prime[--r]=c; else ++c;
/* build |lambda_prime| from edges */
for (j=0; j<s; ++j) mu[i++]=1; /* extend |mu| with |s| ones */
wt_ins(mu,n_tableaux(lambda_prime,l),k%2);
for (j=0; j<s; ++j) mu[--i]=0; /* remove the ones again */
}
if (i>0) goto resume;
{ freearr(edge); freearr(mu); }
}
return wt_collect();
}
local simpgrp* simp_type(entry** m, entry n)
{ matrix* adjs=mkmatrix(n,3);
entry** adj=adjs->elm /* |adj[i]| lists up to 3 neighbours of node |i| */
,* norm=mkintarray(3*n) /* norms of roots */
,* valency=&norm[n] /* valencies in Dynkin diagram */
,* p=&valency[n]; /* permutation of |n| */
simpgrp* result;
lie_Index i,j,k, a_val[4]={-1,-1,-1,-1};
/* |a_val[i]| is index of a node of valency |i|, if any */
if (n==0) error("empty input in simp_type\n");
{ for (i=0;i<n;i++) valency[i]=0;
/* |valency[i]| is also index of next slot in |adj[i]| */
for (i=n; --i>=0;)
{ norm[i]= Norm(m[i]); /* where |Norm(x)==Inprod(x,x)/2| */
for (j=i; --j>=0;)
if (Inprod(m[i],m[j])!=0) /* then valencies increase */
{ if (valency[i]>=3 || valency[j]>=3) error ("valency >3 found\n");
adj[i][valency[i]++]=j; adj[j][valency[j]++]=i;
/* update valencies and adjacencies */
}
a_val[valency[i]]=i; /* valency of node |i| is now known */
}
}
if (a_val[3]<0)
{ lie_Index e; /* index of end node (|valency[e]<=1|) */
if (a_val[0]>=0) p[0]=e=a_val[0]; /* must be type $A_1$ */
else
{ if (a_val[1]>=0) p[0]=e=a_val[1]; /* other linear types */
else error("no end node found\n");
{ k=p[1]=adj[e][0]; /* the unique neighbour of node |e| */
for(i=2;i<n;i++) p[i]=k=opposite(p[i-2],k); /* here |k==p[i-1]| */
}
if ( n==2 && norm[p[0]]+2*norm[p[1]]==5
|| n>=3 && norm[p[0]]!=norm[p[1]]
|| n==4 && norm[p[1]]<norm[p[2]]
)
{ for (i=0; i<n-1-i; i++) swap(&p[i],&p[n-1-i]); e=p[0]; }
}
{ entry norm0=norm[p[0]], norm1=norm[p[n-1]];
if (norm0==norm1) result = mksimpgrp('A',n);
else if (norm1==3) result=mksimpgrp('G',2);
else if (norm1==2) result=mksimpgrp('C',n);
else if (norm0!=2) error("I don't recognize this Cartan Type\n");
else if (n==4 && norm[p[2]]==1) result=mksimpgrp('F',4);
else result=mksimpgrp('B',n);
}
}
/* no nodes of valency 3 */
else
{ entry* branch=adj[a_val[3]], end[3], end_count=0;
for (j=2; j>=0; j--)
if (valency[branch[j]]==1) end[end_count++]=branch[j];
if (end_count>1)
{ p[n-1]=end[1]; p[n-2]=end[0]; p[n-3]=a_val[3];
k=p[n-4]=branch[0]+branch[1]+branch[2]-p[n-1]-p[n-2];
/* the remaining branch */
for(i=n-5; i>=0; i--)
{ if (valency[k]!=2) error("unlinear Dn tail.\n");
p[i]=k=opposite(p[i+2],k);
}
result=mksimpgrp('D',n);
}
else if (end_count==1)
{ p[3]=a_val[3]; p[1]=end[0];
for (j=2; j>=0; j--)
if (valency[branch[j]]==2)
if (valency[opposite(a_val[3],branch[j])]==1) break;
if (j<0) error("type E not recognised\n");
p[2]=branch[j]; p[0]=opposite(p[3],p[2]);
p[4]=k=branch[0]+branch[1]+branch[2]-p[1]-p[2]; /* remaining branch */
for(i=5;i<n;i++)
{ if (valency[k]!=2) error("wrong type E system.\n");
p[i]=k=opposite(p[i-2],k);
}
result=mksimpgrp('E',n);
}
else error("no end node adjacent to valency 3 node\n");
}
for (i=0; i<n; i++) if (p[i]>=0) /* then |p[i]| starts an untreated cycle */
{ entry* mi=m[j=i]; /* record beginning of cycle */
while (p[j]!=i)
{ k=j; j=p[j]; m[k]=m[j]; p[k]= -1; }
/* assign |m[j]=m[p[j]]| and advance */
m[j]=mi; p[j]= -1; /* close the cycle */
}
freemem(adjs); freearr(norm);
return result;
}