#16126: Introduce a class for generalized Coxeter graphs
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Reporter: jipilab | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.2
Component: combinatorics | Keywords: coxeter, graphs, days57
Merged in: | Authors:
Reviewers: | Report Upstream: N/A
Work issues: | Branch:
Commit: | Dependencies:
Stopgaps: |
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Coxeter graphs are undirected graph without multiedges with possibly
labeled edges within the set {4,5,...,oo}. Coxeter graphs encode Coxeter
systems. A Coxeter system is a group presented with a special set of
generators.
To represent a Coxeter group, one can use the canonical geometric
representation, see Chapter 5 of Humphreys for instance. In this
representation, the group acts faithfully on a vector space and preserves
a canonical bilinear form.
For infinite Coxeter groups, it is useful to allow a more general bilinear
form, which still gives a faithful representation and has the advantage
that every reflection subgroup is represented as a general geometric
representation.
These more general representations can be encoded into a generalized
Coxeter graph where labels "oo" can be replaced by any real number $c <=
-1$. The case $c=-1$ represents the canonical choice, in this case one can
keep the label "oo".
The class for generalized Coxeter graphs will be based on the class
DynkinDiagram and adapted for its purposes.
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Ticket URL: <http://trac.sagemath.org/ticket/16126>
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