atlas
0.6

Right multiplication action of simple reflections on a Weyl group modulo (to the left) a maximal parabolic subgroup. More...
#include <weyl.h>
Public Member Functions  
Transducer ()  
Transducer (const int_Matrix &, size_t)  
~Transducer ()  
unsigned long  length (WeylElt::EltPiece x) const 
Length of minimal coset representative x. More...  
unsigned long  maxlength () const 
Maximal length of minimal coset representatives. More...  
Generator  out (WeylElt::EltPiece x, Generator s) const 
Simple reflection t (strictly preceding s) so that xs = tx, if any. More...  
WeylElt::EltPiece  shift (WeylElt::EltPiece x, Generator s) const 
Right coset x' defined by x' = xs. More...  
unsigned long  size () const 
Number of cosets W_{r1}\W_r. More...  
const WeylWord &  wordPiece (WeylElt::EltPiece x) const 
Reduced decomposition in W (or W_r) of minimal coset representative x. More...  
Private Attributes  
std::vector< ShiftRow >  d_shift 
Right multiplication by $s_j$ gives transition i > d_shift[i][j]. More...  
std::vector< OutRow >  d_out 
If d_shift[i][j]==i then $s_j$ transduces in state $i$ to $s_k$ with $k=d_out[i][j]$ (otherwise d_out[i][j]==UndefGenerator). More...  
std::vector< unsigned long >  d_length 
Lengths of the minimal coset representatives $x_i$. More...  
std::vector< WeylWord >  d_piece 
Reduced expressions of the minimal coset representatives. More...  
Right multiplication action of simple reflections on a Weyl group modulo (to the left) a maximal parabolic subgroup.
In the notation from the description of the class WeylGroup, there will be one Transducer object for each parabolic subquotient W_{r1}\W_r. List the shortest length coset representatives for this subquotient as x_0,...,x_{N_r1}. Recall that the simple roots were ordered to guarantee that N_r1 fits in an unsigned char, so each coset representative can be indexed by an unsigned char. We wish to compute the product x_i.s_j for j between 1 and r. The key theoretical fact about multiplication is that there are two mutually exclusive possibilities:
x_i.s_j = x_{i'} (some i' ne i)
OR
x_i.s_j = s_k.x_i (some k < r).
The first possibility is called transition and the second transduction. (Confusingly Fokko's 1999 paper interchanges these terms at their definition, but their usual meaning and the sequel makes clear that this was an error).
The Transducer has tables to describe the two cases. the first table d_shift describes the transistions, namely d_shift[i][j]==i' in the first case; the cases that are transductions can be distinguished from these by the fact that d_shift[i][j]==i. In these cases, the value k emitted by the transduction is stored in d_out[i][j], which otherwise contains the value UndefGenerator

inline 
atlas::weyl::Transducer::Transducer  (  const int_Matrix &  c, 
size_t  r  
) 
$ of length $m2$ 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 $m2$ times, or $m2$ times followed by an upward step; then $xst$ goes up.

inline 

inline 
Length of minimal coset representative x.

inline 
Maximal length of minimal coset representatives.
This is the number of positive roots for the Levi subgroup L_r, minus the number of positive roots for L_{r1}.

inline 
Simple reflection t (strictly preceding s) so that xs = tx, if any.
In case of a transition, this returns UndefGenerator.

inline 
Right coset x' defined by x' = xs.
When x' is not equal to s, this is an equality of minimal coset representatives. When x'=x, the equation for minimal coset representatives is out(x,s).x = x.s.

inline 
Number of cosets W_{r1}\W_r.

inline 
Reduced decomposition in W (or W_r) of minimal coset representative x.

private 
Lengths of the minimal coset representatives $x_i$.

private 
If d_shift[i][j]==i then $s_j$ transduces in state $i$ to $s_k$ with $k=d_out[i][j]$ (otherwise d_out[i][j]==UndefGenerator).
In this case $x_i.s_j = s_k.x_i$, so the state $i$ remains unchanged.

private 
Reduced expressions of the minimal coset representatives.

private 
Right multiplication by $s_j$ gives transition i > d_shift[i][j].