#18645: Add some methods to CartanMatrix
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Reporter: jonathan.judge | Owner: jonathan.judge
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.8
Component: combinatorics | Resolution:
Keywords: days65 | Merged in:
Authors: | Reviewers: Ben Salisbury
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/jonathan.judge/add_some_methods_to_cartanmatrix|
7e17029e1388141ba3cce2f160adc296afb18fcb
Dependencies: | Stopgaps:
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Comment (by jipilab):
Hi all,
Hmm. I also had thoughts about that recently when classifying finite and
affine types. Here are some thoughts, suggestions, feel free to
critize/ignore...
Have a look at #17798, there is a function
"recognize_coxeter_type_from_matrix".
I usually use the following naming convention for Finite and Affine:
-Finite types correspond to positive definite bilinear forms (reducible or
not). For example, A_1 x A_1 should return True to "is_finite".
-Affine types are not finite but the corresponding bilinear is semi-
positive definite. The group acts affinely on a Euclidean space. Note the
"potentiality" here in the definition. Some may require that the dimension
of the kernel of the bilinear be 1-dimensional, so that an affine type
should consist of only one affine component and possibly finite
components.
I like to refer to finite and affine types as level 0 types (see below).
Hyperbolic (Bourbaki and Humphreys): The bilinear form is regular (trivial
kernel) but has exactly one negative eigenvalue. Moreover, it is called
compact if the fundamental cone is compact in the hyperbolic space. It is
called finite volume if the volume of the cone is finite volume in the
hyperbolic space.
+'s: this is classical nomenclature.
-'s: there are way more matrices that are regular and have exactly one
negative eigenvalue.
The only reason I found in the litterature about naming these groups
hyperbolic was that they are the ones that one can easily deal with (at
the time). I believe this fact to be obsolete nowadays.
There is another naming convention which says that hyperbolic is exactly
when the bilinear form is regular and has exactly one eigenvalue. This was
adopted by Vinberg and Maxwell for example in the 70's and 80's. But this
leads to confusion.
Another convention is when the bilinear form has exactly one negative
eigenvalue and a non-zero kernel: weakly hyperbolic. IMHO, this "weakly"
is misleading.
Have a look at Remark 2.2 of http://arxiv.org/abs/1310.8608 where there
are references about this.
==============
To clarify all this, I like to use the notion of level and strictness that
reveals a lot of structure.
A Coxeter matrix or Coxeter graph is of level 0 if it is finite or affine.
A Coxeter matrix or Coxeter graph is of "level =<r" if removing any set of
r generators leaves a finite or affine type graph/matrix.
A Coxeter matrix or Coxeter graph is of level (exactly) r if it is not of
level <=r-1 but of level <=r.
A Coxeter matrix or Coxeter graph is strict if removing the r vertices
always leaves a finite type.
The hyperbolic (compact and non compact) are the graphs of level 1. The
compact hyperbolic types are exactly the strict level 1.
The level 2 graphs have signature (n-1,1,0). This was proved by Maxwell
and his enumeration was corrected in the above paper.
Instead of using hyperbolic I prefer to use the term Lorentzian to reflect
the fact that the group acts not only on the hyperbolic space (which is
part of a Lorentz space) but also on the outside of it. Note that the
Lorentzian represent a potentiality. Lorentzian types are when the
signature of the bilinear is (n-1,1,0). When the kernel is non-zero, i.e.
signature (m,1,k) I would use the term degenerate Lorentzian instead of
weakly hyperbolic.
Sorry for the long text... Anyhow, this is more or less the picture as for
Coxeter matrices and types.
As for the methods: very good to have them there!
--
Ticket URL: <http://trac.sagemath.org/ticket/18645#comment:11>
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