#20402: Make subword complexes compatible with real reflection groups
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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 |
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Comment (by tscrim):
Replying to [comment:46 stumpc5]:
> Replying to [comment:45 tscrim]:
> Of course one could say that k^n^ has an implicit basis in which the
above matrix represents a linear map, but that is not quite adequate since
k^n^ comes with a canonical standard basis, so there is no choice of an
implicit basis.
There is a standard basis, but it is not canonical.
> > That is not how a `CoxeterMatrixGroup` or a `WeylGroup` is defined. It
is given precisely by that choice of representation/matrix, so in a way,
it is a Coxeter system along with a specified representation.
>
> That's right, and I am fine with arguing that this action of the group
should be used when doing {{{w*v}}}. But there are still other spaces
involved on which the group also acts, so if you do not specify which
space {{{v}}} lives in one has to **implicitly** assume it to be this
representation.
This is no more implicit than the relationship with the chosen basis of a
matrix and a vector. I still contend that by fixing a representation, we
now have a canonical basis for `V` (and we implicitly assume `v` is an
element of `V`). For good reasons, we can't tell if `v` is to be
considered an element in the representation or some other (isomorphic)
vector space.
> > In case there is any ambiguity, I'm also only really advocating for
using `*` in the real reflection group setting.
>
> My proposition for reflection groups at the moment is
>
> * have a method {{{.action(vec, side="left", on_space="primal")}}} that
does the appropriate action, and
> * have {{{._act_on_(vec, self_on_left)}}} defined as {{{.action(vec,
side=side, on_space="primal")}}} where the side is set depending on
self_on_left being {{{True}}} or {{{False}}}.
That is good with me.
--
Ticket URL: <http://trac.sagemath.org/ticket/20402#comment:48>
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