#20445: Iteration through Coxeter groups
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Reporter: stumpc5 | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-7.2
Component: combinatorics | Resolution:
Keywords: | Merged in:
Authors: | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: | Stopgaps:
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Description changed by stumpc5:
Old description:
> The algorithm in chevie is very different from the algorithm we currently
> have:
> {{{
> ForEachElement:=function(W,f)local l,g;
> if not IsFinite(W) then Error("only for finite Coxeter groups");
> elif W.nbGeneratingReflections=0 then f(W.identity);return;
> fi;
> l:=List([0..W.nbGeneratingReflections],i->
> ReflectionSubgroup(W,W.reflectionsLabels{[1..i]}));
> l:=List([1..Length(l)-1],i->ReducedRightCosetRepresentatives(l[i+1],l[i]));
> g:=function(x,v)local y;
> if Length(v)=0 then f(x);
> else for y in v[1] do g(x*y,v{[2..Length(v)]});od;
> fi;
> end;
> g(W.identity,l);
> end;
> }}}
>
> We should also implement that algorithm (maybe even with hard-coded coset
> representatives in the finite case) so that we can see if this is indeed
> faster.
New description:
The algorithm in chevie is very different from the algorithm we currently
have:
{{{
ForEachElement:=function(W,f)local l,g;
if not IsFinite(W) then Error("only for finite Coxeter groups");
elif W.nbGeneratingReflections=0 then f(W.identity);return;
fi;
l:=List([0..W.nbGeneratingReflections],i->
ReflectionSubgroup(W,W.reflectionsLabels{[1..i]}));
l:=List([1..Length(l)-1],i->ReducedRightCosetRepresentatives(l[i+1],l[i]));
g:=function(x,v)local y;
if Length(v)=0 then f(x);
else for y in v[1] do g(x*y,v{[2..Length(v)]});od;
fi;
end;
g(W.identity,l);
end;
}}}
We should also implement that algorithm (maybe even with hard-coded coset
representatives in the finite case) so that we can see if this is indeed
faster.
The two needed methods are
{{{
#############################################################################
##
#F ReducedInRightCoset( <W> , <w> ) . . . . . reduced element in coset
W.w
##
## w is an automorphism of a parent Coxeter group of W.
## 'ReducedInRightCoset' returns the unique element in the right
## coset W.w which is W-reduced.
##
AbsCoxOps.ReducedInRightCoset:=function(W,w)local i;
while true do
i := FirstLeftDescending(W, w);
if i=false then return w; fi;
w := W.reflections[i] * w;
od;
end;
#############################################################################
##
#F ReducedRightCosetRepresentatives( <W>, <H> ) . . . . . . . . . . . .
.
#F . . . . . . . . . . . . . . . distinguished right coset
representatives
##
## 'ReducedRightCosetRepresentatives' returns a list of reduced words in
the
## Coxeter group <W> that are distinguished right coset representatives
for
## the right cosets H\W where H is a reflection subgroup of W.
##
ReducedRightCosetRepresentatives:=function(W, H)local res, totest, new;
totest:=Set([W.identity]);
res:=Set([W.identity]);
repeat
new:=Concatenation(List(totest,w->List(
W.reflections{W.generatingReflections},s->ReducedInRightCoset(H,
w*s))));
UniteSet(res,totest);
totest:=Difference(new,res);
until Length(totest)=0;
InfoChevie2("#I nb. of cosets: ",Length(res),"\n");
SortBy(res,x->CoxeterLength(W,x));
return res;
end;
}}}
--
--
Ticket URL: <http://trac.sagemath.org/ticket/20445#comment:1>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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