#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:44 stumpc5]:
> > 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.
>
> I'd say that multiplying a matrix {{{M}}} with a vector {{{v}}} (as a
tuple in distinction to vector space elements) is well-defined and
completely independent on bases. So that's totally fine to do as
{{{M*v}}}.
No, that is not true except in the most abstract sense of a linear
morphism and an element of a vector space. A matrix comes with an implicit
choice of a basis, and when we write vectors as tuples, there is an
implicit choice of a basis as well. For example, if `M` is a
diagonalizable matrix acting on a vector space `V`, we can find some basis
such that `M` can be expressed as acting on scalars on that basis. While
the diagonal matrix and `M` are different matrices, they define the
''same'' linear automorphism on `V`, and hence, have the same action on a
vector. Yet, to do the computation, we need to express the vector in terms
of our basis.
> Only if the matrix is representing a linear map and the tuple represents
a vector space element there is an assumption on both being represented in
the same basis. When you now represent a Coxeter group element as a matrix
you make a choice of basis (which might be the root basis, its dual, some
linearly independent vectors of an "ambient space", or any other) This is
now what you stick this matrix to, but when you do {{{w*v}}}, one does not
specify the matrix that represents {{{w}}} in some basis, so this
operation is ambiguous, as it didn't specify on which space {{{w}}} acts
here (i.e., it is not clear in which basis you represent {{{w}}} as a
matrix for that action.
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.
Also, any `v` (implicitly) carries with it a vector space (not up to
isomorphism), and we are checking to see if the Coxeter group has a
defined action on that vector space. An alternative viewpoint is if `v` is
a Sage vector, in a way, we are only considering that element up to
isomorphism, so we have lost some information (but again, that is if we
don't fix a representation which makes a canonical choice of basis for the
vector space). In contrast, for `v` being an element of the
root/weight/ambient space, we have an explicit basis and we have defined
where the roots are in these, so the action is well-defined and explicit.
> For {{{ReflectionGroup}}} I just thought that the best is to have the
method {{{.action}}} to take two parameters {{{side}}} being {{{"left"}}}
or {{{"right"}}} and {{{on_space}}} being {{{"primal"}}} or {{{"dual"}}}.
>
> Beside that, I am okay with providing the coercion to act on the primal,
but am also okay with removing it again (I am slightly in favour of
keeping it though).
In case there is any ambiguity, I'm also only really advocating for using
`*` in the real reflection group setting. Moreover, I think we should
avoid using Sage vectors for the roots (at least in the generic parents)
because there is some ambiguity about their basis.
--
Ticket URL: <http://trac.sagemath.org/ticket/20402#comment:45>
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