/*
Synopsis: outputs information about the strong real forms of G.
Explanation: the inverse image in \X of a class of weak real forms of G is
of the form Z.\X(z), where z is an admissible value for x^2 for a strong real
form of that class; so there is associated to the class of weak real forms
a coset (1+\delta)(Z).z in Z^\delta. All the various \X(z) for all possible
choices of z are isomorphic by Z-translation. Therefore it is enough to
describe the combinatorial structure of one of them, for each class of weak
real forms. This is a finite problem in all cases.
We output the orbits of W_im in X(z), which correspond to the various strong
real forms; we label them with the corresponding weak real form.
*/
std::ostream& printStrongReal(std::ostream& strm,
InnerClass& G_C,
const output::FormNumberMap& rfi,
size_t cn)
{
const CartanClass& cc = G_C.cartan(cn);
const RealFormNbrList& rfl = G_C.realFormLabels(cn);
size_t n = cc.numRealFormClasses();
if (n>1)
strm << "there are " << n << " real form classes:\n" << std::endl;
for (cartanclass::square_class csc=0; csc<n; ++csc)
{
// print information about the square of real forms, in center
{
RealFormNbr wrf=cc.fiber().realFormPartition().classRep(csc);
RealFormNbr fund_wrf= rfl[wrf]; // lift weak real form to |fund|
cartanclass::square_class f_csc=G_C.xi_square(fund_wrf);
// having the square class number of the fundamental fiber, get grading
RatCoweight coch = some_coch(G_C,f_csc);
Grading base_grading = grading_of_simples(G_C,coch);
RatWeight z (G_C.rank());
for (Grading::iterator it=base_grading.begin(); it(); ++it)
z += G_C.rootDatum().fundamental_coweight(*it);
Ratvec_Numer_t& zn = z.numerator();
for (size_t i=0; i<z.size(); ++i)
zn[i]=arithmetic::remainder(zn[i],z.denominator());
strm << "class #" << f_csc
<< ", possible square: exp(2i\\pi(" << z << "))" << std::endl;
}
const Partition& pi = cc.fiber_partition(csc);
unsigned long c = 0;
for (Partition::iterator i(pi); i(); ++i,++c)
{
std::ostringstream os;
RealFormNbr rf = rfl[cc.toWeakReal(c,csc)];
os << "real form #" << rfi.out(rf) << ": ";
basic_io::seqPrint(os,i->first,i->second,",","[","]")
<< " (" << i->second - i->first << ")" << std::endl;
ioutils::foldLine(strm,os.str(),"",",");
}
if (n>1)
strm << std::endl;
}
return strm;
}
bool is_central(const WeightList& alpha, const TorusElement& t)
{
RatWeight rw = t.as_Qmod2Z();
arithmetic::Numer_t d = 2*rw.denominator();
for (weyl::Generator s=0; s<alpha.size(); ++s)
if (alpha[s].dot(rw.numerator())%d != 0) // see if division is exact
return false;
return true;
}