#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 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}}}.
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 {{{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).
--
Ticket URL: <http://trac.sagemath.org/ticket/20402#comment:44>
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