poly* Plethysm(entry* lambda,_index l,_index n,poly* p)
{ if (n==0) return poly_one(Lierank(grp)); else if (n==1) return p;
{ _index i,j;
poly* sum= poly_null(Lierank(grp)),**adams=alloc_array(poly*,n+1);
poly* chi_lambda=MN_char(lambda,l);
for (i=1; i<=n; ++i) { adams[i]=Adams(i,p); setshared(adams[i]); }
for (i=0;i<chi_lambda->nrows;i++)
{ entry* mu=chi_lambda->elm[i]; poly* prod=adams[mu[0]],*t;
for (j=1; j<n && mu[j]>0; ++j)
{ t=prod; prod=Tensor(t,adams[mu[j]]); freepol(t); }
sum= Addmul_pol_pol_bin(sum,prod,mult(chi_lambda->coef[i],Classord(mu,n)));
}
freemem(chi_lambda);
setshared(p); /* protect |p|; it coincides with |adams[1]| */
for (i=1; i<=n; ++i)
{ clrshared(adams[i]); freepol(adams[i]); } freearr(adams);
clrshared(p);
{ bigint* fac_n=fac(n); setshared(fac_n); /* used repeatedly */
for (i=0; i<sum->nrows; ++i)
{ bigint** cc= &sum->coef[i]
,* c= (clrshared(*cc),isshared(*cc)) ? copybigint(*cc,NULL) : *cc;
*cc=divq(c,fac_n); setshared(*cc);
if (c->size!=0) error("Internal error (plethysm).\n"); else freemem(c);
}
clrshared(fac_n); freemem(fac_n);
}
return sum;
}
}
poly* SAtensor(boolean alt,_index m,poly* p)
{ _index n,r=Lierank(grp); poly** adams,** q,* result;
if (m==0) return poly_one(r); else if (m==1) return p;
adams=alloc_array(poly*,m+1);
for (n=1; n<=m; ++n) adams[n]=Adams(n,p);
q=alloc_array(poly*,m+1);
q[0]=poly_one(r);
for (n=1; n<=m; ++n)
{
{ _index i; q[n]=Tensor(p,q[n-1]); /* the initial term of the summation */
for (i=2; i<=n; ++i) q[n] =
Add_pol_pol(q[n],Tensor(adams[i],q[n-i]),alt&&i%2==0);
}
{ _index i; bigint* big_n=entry2bigint(n); setshared(big_n);
for (i=0; i<q[n]->nrows; ++i)
{ bigint** cc= &q[n]->coef[i]
,* c= (clrshared(*cc),isshared(*cc)) ? copybigint(*cc,NULL) : *cc;
*cc=divq(c,big_n); setshared(*cc);
{ if (c->size != 0)
error("Internal error (SAtensor): remainder from %ld.\n" ,(long)n);
freemem(c);
}
}
clrshared(big_n); freemem(big_n);
}
}
result=q[m];
{ for (n=1; n<=m; ++n) freepol(adams[n]); } freearr(adams);
{ for (n=0; n<m; ++n) freepol(q[n]); } freearr(q);
return result;
}
void wt_ins(entry* wt, bigint* c, boolean neg)
{ if (c->size==0) {
freemem(c);
return;
}
{ lie_Index i=searchterm(sorted,wt);
if (i>=0)
{ clrshared(sorted->coef[i]);
sorted->coef[i]= (neg ? sub : add)(sorted->coef[i],c);
setshared(sorted->coef[i]);
}
else
{ poly** acc= neg ? &neg_acc : &pos_acc;
lie_Index i=(*acc)->nrows;
if (i==(*acc)->rowsize)
{ sorted=Add_pol_pol(sorted,*acc,neg);
*acc=mkpoly(Max(sorted->nrows,ACCMIN),sorted->ncols);
i=0;
}
copyrow(wt,(*acc)->elm[i],sorted->ncols);
(*acc)->coef[i++]=c;
setshared(c);
(*acc)->nrows=i;
}
}
}
poly* From_Part_p (poly* p)
{ _index i,j,n_rows=p->nrows,n=p->ncols; poly* result=mkpoly(n_rows,n-1);
entry** lambda=p->elm; entry** res=result->elm;
for (i=0; i<n_rows; ++i)
{ result->coef[i]=p->coef[i]; setshared(p->coef[i]); /* copy coefficient */
for (j=0; j<n-1; ++j) res[i][j]=lambda[i][j]-lambda[i][j+1];
}
return Reduce_pol(result);
}
poly* To_Part_p (poly* p)
{ _index i,n_rows=p->nrows,n=p->ncols; entry** wt=p->elm;
poly* result=mkpoly(n_rows,n+1); entry** lambda=result->elm;
for (i=0; i<n_rows; ++i)
{ _index j=n; entry sum=0;
result->coef[i]=p->coef[i]; setshared(p->coef[i]);
while (lambda[i][j]=sum, --j>=0) sum+=wt[i][j];
}
return Reduce_pol(result);
}
poly *pol_mul_pol_mat(poly *a, matrix *b) {
poly *m;
int i;
if (a->ncols != b->nrows)
error("Number variables arg1 unequal number of rows arg2 .\n");
m = mkpoly(a->nrows,b->ncols);
mulmatmatelm(a->elm,b->elm,m->elm,a->nrows,a->ncols,b->ncols);
for (i=0;i<m->nrows;i++) {
m->coef[i] = a->coef[i];
setshared(m->coef[i]);
}
return m;
}
poly* Reduce_pol(poly* p)
{ entry** expon=p->elm; bigint** coef=p->coef; lie_Index t=0,f=0,len=p->ncols;
heap_sort_p(p,cmpfn);
/* don't exclude cases~$<2$: we must catch $0$-polynomials */
while (++f<p->nrows)
if (coef[f]->size==0) clrshared(coef[f]); /* drop term with zero coef */
else if (eqrow(expon[f],expon[t],len)) /* equal exponents: add coef's */
{ clrshared(coef[t]); clrshared(coef[f]);
coef[t]=add(coef[t],coef[f]); setshared(coef[t]);
}
else /* now term at f replaces one at t as discriminating term */
{ if (coef[t]->size) t++; else clrshared(coef[t]); /* keep if nonzero */
swap_terms(expon,coef,t,f); /* move term, preserve row separateness */
}
if (p->nrows!=0)
/* |p| mights have no terms at all (e.g. from |alt_dom|). */
if (coef[t]->size) t++; else clrshared(coef[t]); /* handle final term */
else *coef=copybigint(null,NULL); /* safer not to introduce aliasing */
if ((p->nrows=t)==0) /* then must keep last term; coef is cleared */
{ lie_Index i; p->nrows=1; setshared(*coef); /* |*coef| was |0| but not shared */
for (i=0; i<len; i++) expon[0][i]=0; /* clear first exponent as well */
}
setsorted(p); return p;
}
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 */
}
poly* Adjoint(object grp)
{ _index i,j,r=Lierank(grp)
,n=type_of(grp)==SIMPGRP ? 1: grp->g.ncomp+(grp->g.toraldim!=0);
poly* adj= mkpoly(n,r);
for (i=0; i<n; ++i)
{ adj->coef[i]=one; for (j=0; j<r; ++j) adj->elm[i][j]=0; }
if (type_of(grp)==SIMPGRP) set_simp_adjoint(adj->elm[0],&grp->s);
else
{ _index offs=0; simpgrp* g;
for (i=0; i<grp->g.ncomp; offs+=g->lierank,++i)
set_simp_adjoint(&adj->elm[i][offs],g=Liecomp(grp,i));
if (grp->g.toraldim!=0)
{ adj->coef[i]=entry2bigint(grp->g.toraldim);
setshared(adj->coef[i]);
}
}
return adj;
}
poly* Worbit_p(poly* p)
{ lie_Index i,k=0,l=0,r=p->ncols; poly* result; entry** res;
p=copypoly(p); for (i=0; i<p->nrows; ++i) make_dominant(p->elm[i]);
Reduce_pol(p);
for (i=0; i<p->nrows; ++i)
if ((l += bigint2entry(Orbitsize(p->elm[i])))<0)
error ("That's too large an orbit");
result=mkpoly(l,p->ncols); res=result->elm;
for (i=0; i<p->nrows; ++i)
{ lie_Index j; matrix* orbit=Weyl_orbit(p->elm[i],NULL); entry** x=orbit->elm;
for (j=0; j<orbit->nrows; ++j)
{ result->coef[k]=p->coef[i]; setshared(p->coef[i]);
copyrow(*x++,res[k++],r);
}
freemem(orbit);
}
assert(k==result->nrows);
return result; /* not sorted, but rows are unique */
}
