Constructs subquotient #r for the Coxeter matrix c.
This uses the Coxeter matrix only up to index r. In fact we can behave as if generator |r| is the final one, since we ignore any higher ones for now.
Precondition : c is a _normalized_ Coxeter matrix (meaning that all the parabolic subquotients W_{r-1}\W_r are small enough to fit in an unsigned char); and r is < rank(c);
Algorithm :
The algorithm is a version of my favorite bootstrapping procedure for the construction of Weyl groups and parabolic quotients. We start with a partially defined automaton containing only one element, and for which all shifts by the final generator $r$ are not yet defined (generators $i<r$ give transduction of $i$). At each point in time, all shifts for all elements in the automaton that do _not_ take the length up are defined; and we maintain a queue of elements that may have as yet undefined shifts.
We start up with one element in the automaton, with just one undefined shift, the one by r. Then run through the elements $x$ of the automaton in order of generation (which will also be in ShortLex order), and for each as yet undefined shift of $x$ by $s$ :
- add a new element $xs$ (this increases the size of the automaton, during the loop!). At this point it is sure that this element is really new, and that its normal form is obtained by adding $s$ to that of $x$
- then find all other elements $x'$ already in the automaton and $t$ such that $xs==x't$ (so $x't$ gives a non-canonical but reduced expression for $xs$). Do this by trying generators
: if $xst$ goes down (has the same length as $x$) then $x'==xst$ gives such a case. The trick for this is to look at the orbit of x under the dihedral group <s,t>. In the full group, this has necessarily cardinality 2m, with m = m(s,t) the coefficient in the Coxeter matrix, and $l(xst)==l(x)$ iff $xs$ is the unique elt. of maximal length in the orbit, hence to have this $x$ must goes down $m-1$ times when applying successively $t$, $s$, $t$, ... In the parabolic quotient, the orbit of the dihedral group (which is not reduced to a point) can either have cardinality $2m$ or cardinality $m$, and in the latter case it is a string with $m-1$ steps between the bottom and the top, with a stationary step at either extreme (to see this, note that on one hand each step up in the full group gives a step in the quotient that is either up or stationary, while on the other hand a stationary step in the quotient causes then next step to be the reverse of the previous one). So we have one of the following cases: (1) $x$ goes down $m-1$ times; then the image of the orbit has $2m$ elements and $xs==x't$ for $x'=x.(ts)^(m-1)$. (2) $x$ goes down $m-2$ times to some element $a$ and is then stationary (if 
is the next to apply, then $a.s'=g.a$ for some generator 
); then the orbit has $m$ elements, and if $v$ is the alternating word in 
of length $m-2$ not starting with $s'$, so that $a.v=x$, one has $v.st=s'vs$ whence $x.st=a.v.st=a.s'vs=g.a.vs=g.xs$ so that $xs$ has a transduction for $t$ that outputs the generator $g$. (3) either $x$ goes down less than $m-2$ times, or $m-2$ times followed by an upward step; then $xst$ goes up.
Definition at line 781 of file weyl.cpp.
References d_length, d_out, d_piece, d_shift, atlas::dihedralMin(), atlas::dihedralShift(), atlas::weyl::EltPiece, atlas::weyl::Generator, atlas::latticetypes::LatticeCoeff, atlas::latticetypes::LatticeMatrix, atlas::weyl::OutRow, atlas::weyl::ShiftRow, and atlas::weyl::WeylWord.
