#20402: Make subword complexes compatible with real reflection groups
-------------------------------------+-------------------------------------
Reporter: stumpc5 | Owner:
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-7.2
Component: combinatorics | Resolution:
Keywords: reflection group, | Merged in:
coxeter group, subword complex, | Reviewers:
days80 | Work issues:
Authors: Christian Stump | Commit:
Report Upstream: N/A | 295d784db0ae24bed97ed7b4d3777df9dbd652c2
Branch: u/stumpc5/20402 | Stopgaps:
Dependencies: #11187 |
-------------------------------------+-------------------------------------
Comment (by tscrim):
Replying to [comment:42 stumpc5]:
> Replying to [comment:38 tscrim]:
> > my interpretation of Christian's first comment was this was a property
of the realization of the group, not of the group itself. So the group's
`is_crystallographic` should just call that of its Coxeter type/matrix.
>
> Yes, for real groups I only wanted the `is_crystallographic` of the
group to call the one of the Coxeter type/matrix. But complex reflection
groups can still be crystallographic (though they only are if they are
real and crystallographic), so they deserve the method
{{{is_crystallographic}}} without having a Coxeter type/Coxeter matrix
attached to them.
Ah, I see. This good.
> > So I see no ambiguity, and in fact, I think the Weyl groups are very
explicit about this.
>
> But what if {{{w}}} is an element of the group and {{{v}}} is a *Sage
vector*. Does it represent an element in the vector space spanned the
simple roots, or by the simple coroots, or by the fundamental weights, or
by the fundamental coweights? Or does that not matter since the actions
are all the same?
Sage's vectors exists expressed in some canonical basis, and by doing
matrix multiplication, there is an implicit assumption that these bases
are the same. In a way, what you are advocating for is that we need to
explicitly specify the matrix for any matrix `m` and `v` and check those
are the same anytime we write `m * v`. The various subclasses of
`CombinatorialFreeModule` does this by saying the (parent) class
corresponds to a choice of basis, but this is not, and really can never
be, the case for Sage's vectors, where there is an assumption of a
canonical basis. In fact, because these matrices are representations of a
vector space given in terms of a basis, that basis is canonical as far as
Sage's vectors are concerned. (I hope that makes some sense...)
--
Ticket URL: <http://trac.sagemath.org/ticket/20402#comment:43>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
--
You received this message because you are subscribed to the Google Groups
"sage-trac" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at https://groups.google.com/group/sage-trac.
For more options, visit https://groups.google.com/d/optout.
