Implement Coxeter groups in their geometric representation

[nthiery]

The root system / coxeter group code is designed from the ground up to allow for this extension.

Steps:

• Double check CartanType(["H",3]).coxeter_diagram() and friends

• Given a coxeter diagram, construct the dynkin diagram g
  corresponding to the geometric representation; most of the time,
  this will involve roots of unity, and require e.g. a cyclotomic
  field (see also Implement the universal cyclotomic field, using Zumbroich basis #8327)

• Make sure that L = RootSystem(g).root_space() accepts such a diagram

• Make sure that WeylGroup(L) accepts such a root space

• Fix all the interfaces to properly reflect the generalization
  (e.g. WeylGroup above should really be CoxeterGroup).

Apply:

• attachment: trac_9290-geometric_coxeter_groups-ts.patch
• attachment: trac-9290-review.patch

Depends on #15204

CC: @sagetrac-sage-combinat

Component: combinatorics

Keywords: coxeter

Author: Travis Scrimshaw

Reviewer: Frédéric Chapoton

Merged: sage-5.13.beta2

Issue created by migration from https://trac.sagemath.org/ticket/9290

[nthiery]

comment:1

Partially depends on #8237

[nthiery]

Description changed:

--- 
+++ 
@@ -1 +1,18 @@
+The root system / coxeter group code is designed from the ground up to allow for this extension.
 
+Steps:
+
+- Double check `CartanType(["H",3]).coxeter_diagram()` and friends
+
+- Given a coxeter diagram, construct the dynkin diagram `g`
+  corresponding to the geometric representation; most of the time,
+  this will involve roots of unity, and require e.g. a cyclotomic
+  field (see also #8327)
+
+- Make sure that `L = RootSystem(g).root_space()` accepts such a diagram
+
+- Make sure that `WeylGroup(L)` accepts such a root space
+
+- Fix all the interfaces to properly reflect the generalization
+  (e.g. WeylGroup above should really be CoxeterGroup).
+

[fchapoton]

Changed keywords from none to coxeter

[fchapoton]

comment:4

see also the code in the ticket #14816

[tscrim]

comment:5

Here's a patch which should take care of all except the second point since I thought Dynkin diagrams where separate from Coxeter diagrams except when there are 1,2,3, or 4 bonds (and even then, we're ignoring the arrows).

Base patch is from #15137, but now uses the UCF as the default field.

[fchapoton]

comment:7

there are some failing doctests

EDIT: hmm, it rather seems to be a problem with the patchbot..