#26028 closed defect (fixed)
BugFixes and improvements with respect to the bilinear invariant form of classical matrix groups
Reported by:  soehms  Owned by:  

Priority:  major  Milestone:  sage8.4 
Component:  group theory  Keywords:  matrix groups, unitary, symplectic, orthogonla, classical, invariant bilinear form 
Cc:  tscrim  Merged in:  
Authors:  Sebastian Oehms  Reviewers:  Travis Scrimshaw 
Report Upstream:  N/A  Work issues:  
Branch:  331e5cb (Commits)  Commit:  
Dependencies:  #25761  Stopgaps: 
Description
1) Bug with respect to the symplectic groups:
sage: Sp(4, QQ).invariant_form() [0 0 0 1] [0 0 1 0] [0 1 0 0] [1 0 0 0]
This bilinear form isn't alternating. Thus Sp(4, QQ) in fact is an orthogonal group!
2) Bug with respect to the unitary groups:
sage: G = GU(2,2); G General Unitary Group of degree 2 over Finite Field in a of size 2^2 sage: m = matrix(G.base_ring(), 2, 2, (1,1,1,0)); m [1 1] [1 0] sage: m in [g.matrix() for g in G] True sage: G(m) Traceback (most recent call last): ... TypeError: matrix must be unitary
Note, that this bug is not fixed by ticket #25761. The reason for this bug is this:
sage: invariant_form = matrix(G.base_ring(), 2,2, G.gap().InvariantSesquilinearForm()['matrix'].matrix()) sage: invariant_form == G.one().matrix() False
Furthermore, There is no method to obtain the invariant from of a unitary group analogues as for symplectic and orthogonal groups.
sage: GU(3,2).invariant_form() Traceback (most recent call last): ... AttributeError: 'UnitaryMatrixGroup_gap_with_category' object has no attribute 'invariant_form'
3) Bug with respect to the orthogonal group (but not with respect to the invariant form):
sage: GO3_25 = GO(3,25) sage: GO3_25.order() 240 sage: GO3_5 = GO(3,5) sage: GO3_5.order() 240 sage: GO3_5.is_isomorphic(GO3_25) True
Furthermore, it should be possible to define generic classical matrix groups with respect to a user given invariant bilinear form.
Change History (15)
comment:1 Changed 16 months ago by
comment:2 Changed 16 months ago by
Actually, it may not be a problem with gap:
sage: GO(3,4).gap() GO(0,3,2) sage: GO(3,2).gap() GO(0,3,2)
comment:3 Changed 16 months ago by
Yep, I am now sure the issue is with the initializer. In GO
:
if is_FiniteField(ring): cmd = 'GO({0}, {1}, {2})'.format(e, degree, ring.characteristic()) return OrthogonalMatrixGroup_gap(degree, ring, False, name, ltx, cmd)
versus, say Sp
:
cmd = 'Sp({0}, {1})'.format(degree, ring._gap_init_()) return SymplecticMatrixGroup_gap(degree, ring, True, name, ltx, cmd)
I think the former should be in the same as the latter or cardinality()
. With changing characteristic
> cardinality
, I at least get something reasonable:
sage: GO(3,4).gens() ( [ 1 0 0] [1 0 0] [ 0 a 0] [1 1 1] [ 0 0 a + 1], [0 1 0] ) sage: GO(3,4).cardinality() 60 sage: GO(3,8).cardinality() 504
The same change needs to be made for SO
.
comment:4 Changed 16 months ago by
 Branch set to u/soehms/classical_grps_invariant_form26028
comment:5 Changed 16 months ago by
 Commit set to d97fc1e944b707f026f6d0626e5d769a6e2dd066
 Dependencies set to #25761
 Keywords invariant bilinear form added
 Status changed from new to needs_review
Thank your for these suggestions. I've already worked on the ticket but didn't push it up right after creating the ticket since I had build problems after merging with 8.4.beta0. What I do exactly coincides with your suggestions. In detail this is:
The three Bugfixes I implement in
1) method invariant_form
of SymplecticMatrixGroup_generic
.
2) method _check_matrix
of UnitaryMatrixGroup_generic
.
3) new function _OG
in orthogonal.py
.
I add a new method invariant_form
to the classes UnitaryMatrixGroup_generic
and UnitaryMatrixGroup_gap
.
Since the according method of OrthogonalMatrixGroup_generic
was named invarinat_bilinear_form
I add an alias for invariant_form
there. Furthermore, I replace the (in the generic case) identical method invariant_quadratic_form
by an alias, as well.
In order to implement an optional argument to define classical matrix groups with respect to a user given invariant bilinear form, I do the following:
I add the attribute _user_invariant_form_
to the class NamedMatrixGroup_generic
together with an optional argument invariant_form
to set it (taking care of CachedRepresentation?). Furthermore, I add the function normalize_args_invariant_form
(analogous to normalize_args_vectorspace
).
Than, I add the optional argument invariant_form
to the functions GO, SO, GU, SU
and Sp
and transfer them to NamedMatrixGroup_generic
after normalization. Since GO
and SO
(resp. GU
and SU
) use nearly identical code I capsule these in local functions _OG
and _UG
to avoid to much duplicated code.
The behavior with respect to the new argument I realize in the following way:
The argument is checked to define a non singular squarematrix over the given ring according to the given degree. Furthermore, the matrix is checked to by symmetric, hermitian or alternating according to the case of classical group. It is not checked to be positive definite in the case of orthogonal and unitary groups. But it will be printed in the representation string and latex string if it is not. Furthermore, the user given invariant bilinear form will always be printed in these representation strings.
Finally, I add the invariant bilinear form to the error message of the _checke_matrix
method, but in the cases of orthogonal and unitary groups only if it is different from the identity matrix.
Well, this is a lot of stuff. If you think not all of this is necessary, please correct it!
New commits:
591cc0b  Merge branch 'develop' into is_unitary_for_finite_fields25761

2a8dd57  initial implementation

d97fc1e  Merge branch 'classical_grps_invariant_form26028' of /home/sebastian/speedport/sagedev into classical_grps_invariant_form26028

comment:6 Changed 16 months ago by
 Branch changed from u/soehms/classical_grps_invariant_form26028 to public/groups/invariant_form_classical_gps26028
 Commit changed from d97fc1e944b707f026f6d0626e5d769a6e2dd066 to 588b4d37dc33a628839bc1aff55e73567a1e93f0
 Reviewers set to Travis Scrimshaw
I made a bunch of small tweaks. Most of them were doc formatting and convention things. If my changes are good, then you can set a positive review.
New commits:
588b4d3  Did some formatting fixes and other small reviewer tweaks.

comment:7 Changed 16 months ago by
thank you very much, Travis! I agree with all your changes. But there is one problem left:
I saw that you forget to change the doctest concerning the NotImplementedError
to the new text ("... for finite groups is fixed by GAP"). On the other hand all doctests passed!
Why isn't this detected? The doctest after the blank line is completely ignored (you can type any error into it). But if I delete the blankline I'm getting a doctest failure with respect to the former test.
What can we do to ensure the test is performed?
comment:8 Changed 16 months ago by
 Commit changed from 588b4d37dc33a628839bc1aff55e73567a1e93f0 to 331e5cbc6327a37bb9b29af7aba236999e8d72b5
Branch pushed to git repo; I updated commit sha1. New commits:
331e5cb  Fixing doctest issues.

comment:9 Changed 16 months ago by
So the issue was that it was sage foo
not sage: foo
for those tests. So Sage's doctest framework did not pick up that they should actually be tests. Fixed.
comment:10 Changed 16 months ago by
 Status changed from needs_review to positive_review
Sorry, that was stupid!
comment:11 Changed 16 months ago by
I was actually a little surprised the doctest framework didn't catch that, but there are only so many things it can do.
comment:12 Changed 15 months ago by
 Branch changed from public/groups/invariant_form_classical_gps26028 to 331e5cbc6327a37bb9b29af7aba236999e8d72b5
 Resolution set to fixed
 Status changed from positive_review to closed
comment:13 followup: ↓ 14 Changed 9 months ago by
 Commit 331e5cbc6327a37bb9b29af7aba236999e8d72b5 deleted
What's the reason for the try
/except
here?
try: if not invariant_form.is_positive_definite(): inserted_text = 'with respect to non positive definite hermitian form' except: pass
See #27427
comment:14 in reply to: ↑ 13 ; followup: ↓ 15 Changed 9 months ago by
Replying to jdemeyer:
What's the reason for the
try
/except
here?try: if not invariant_form.is_positive_definite(): inserted_text = 'with respect to non positive definite hermitian form' except: passSee #27427
For this:
sage: F.<a> = FiniteField(5^3) sage: M = matrix([[a,1],[a^2,a+2]]) sage: M.is_positive_definite()  ValueError Traceback (most recent call last) <ipythoninput7c019faa7a9b7> in <module>() > 1 M.is_positive_definite() /home/uqtscrim/sage/local/lib/python2.7/sitepackages/sage/matrix/matrix2.pyx in sage.matrix.matrix2.Matrix.is_positive_definite (build/cythonized/sage/matrix/matrix2.c:85468)() 12710 real = False 12711 else: > 12712 raise ValueError("Could not see {} as a subring of the " 12713 "real or complex numbers".format(R)) 12714 ValueError: Could not see Finite Field in a of size 5^3 as a subring of the real or complex numbers
(Sorry Jeroen for missing the bare except:
during my review.)
comment:15 in reply to: ↑ 14 Changed 9 months ago by
Replying to tscrim:
(Sorry Jeroen for missing the bare
except:
during my review.)
And I am sorry for doing that! It was before I learned about that convention, as well. That will not occur any more!
Well, apparently  according to the doc 
Sp
is an orthogonal group:However, to note that for finite fields, it does return the correct result (in part, but GAP does the computation). The generic implementation seems to have forgotten that it should be a skewsymmetric bilinear form. That should be simple to fix.
You seem to have come across a bigger issue for the last one
Contrast with
So the matrix group
GO(3,4)
is quite a bit bigger than6
. I am pretty certain this comes from the groups having the same generators:This is a bug with upstream GAP not creating the generators correctly.
IMO each of these issues should be a separate ticket.