More improvements for braid groups involving permutations, and a 2-generator presentation #35985

enriqueartal · 2023-07-24T16:55:33Z

This PR is a continuation of #35214.

Given an element of a symmetric group, the associated permutation braid can be constructed, added in _element_constructor.
The method _standard_lift_Tietze has been simplified (for a permutation) and now it gives a shortest word for the lift but not necesarily the smallest for the lexicographic order. The former code has been commented.
The method presentation2gens provides a presentation of the braid group with two generators. The general method epimorphism has been adapted to braid groups using this presentation and it is much faster.

📝 Checklist

The title is concise, informative, and self-explanatory.
The description explains in detail what this PR is about.
I have linked a relevant issue or discussion.
I have created tests covering the changes.
I have updated the documentation accordingly.

⌛ Dependencies

Change _standard_lift_Tietze for braids

…tric group

Gcc 13

Libgd reduce deps

src/sage/groups/artin.py

tscrim · 2023-07-26T22:53:44Z

src/sage/groups/artin.py

        OUTPUT:

-        A permutation.
+        An element of the Coxeter group or an element of a symmetric group.


Suggested change

An element of the Coxeter group or an element of a symmetric group.

An element of the Coxeter group ``W``.

src/sage/groups/artin.py

tscrim · 2023-07-26T22:56:25Z

src/sage/groups/artin.py

+        In = coxeter_matrix.index_set()
+        for ii, i in enumerate(In):
+            for j in In[ii+1:]:


Please ignore (IMO stupid) suggestions regarding nonessential code style that moves it further from common mathematical notation.

tscrim · 2023-07-26T22:57:47Z

src/sage/groups/braid.py

+        The image of ``self`` under the natural projection map to ``W``
+        as either a standard permutation or an element of a symmetric group.


Revert this change as the symmetric group is an example of the Coxeter group W and this extra (unnecessary) verbosity makes it more confusing.

tscrim · 2023-07-26T23:05:28Z

src/sage/groups/braid.py

+        - ``isomorphism`` -- boolean (default ``False``). If ``True`` an isomorphism
+          from ``self`` and the isomorphic group and its inverse are provided


Suggested change

- ``isomorphism`` -- boolean (default ``False``). If ``True`` an isomorphism

from ``self`` and the isomorphic group and its inverse are provided

- ``isomorphism`` -- boolean (default: ``False``); if ``True``, then an isomorphism

from ``self`` and the isomorphic group and its inverse is also returned

tscrim · 2023-07-26T23:06:16Z

src/sage/groups/braid.py

@@ -3407,7 +3519,7 @@ def BraidGroup(n=None, names='s'):
    - ``n`` -- integer or ``None`` (default). The number of
      strands. If not specified the ``names`` are counted and the
      group is assumed to have one more strand than generators.
-
+


Please revert this entire change.

I do not understand this entire change. There was an extra + due to my fingers, removed.

Blanklines between inputs are unnecessary and evil IMO. I don't want to add them anywhere, especially where it wasn't before. (It also helps prevent merge conflicts as it is not where you are making changes. Many of your other changes fall under this too, but I don't disapprove of them.)

I reverted both changes, but actually in this file for most classes there is a blank line (not introduced by me)

tscrim · 2023-07-26T23:06:26Z

src/sage/groups/braid.py

@@ -3526,6 +3638,7 @@ class group of the punctured disk.
        sage: A(x1^-1, s1)
        x1*x2^-1*x1^-1
    """
+


Remove; same as before.

src/sage/groups/braid.py

tscrim · 2023-07-26T23:08:57Z

src/sage/groups/braid.py

+        through the presentation of two gens for ``self``
+
+        INPUT:
+
+        - `H` -- Another group
+
+        EXAMPLES::
+
+            sage: B = BraidGroup(5)
+            sage: B.epimorphisms(SymmetricGroup(5))
+            [Generic morphism:
+            From: Braid group on 5 strands
+            To:   Symmetric group of order 5! as a permutation group
+            Defn: s0 |--> (1,5)
+                    s1 |--> (4,5)
+                    s2 |--> (3,4)
+                    s3 |--> (2,3)]


Suggested change

through the presentation of two gens for ``self``

INPUT:

- `H` -- Another group

EXAMPLES::

sage: B = BraidGroup(5)

sage: B.epimorphisms(SymmetricGroup(5))

[Generic morphism:

From: Braid group on 5 strands

To: Symmetric group of order 5! as a permutation group

Defn: s0 |--> (1,5)

s1 |--> (4,5)

s2 |--> (3,4)

s3 |--> (2,3)]

through the :meth:`two generator presentation

<presentation_two_generators>` of ``self``.

INPUT:

- `H` -- another group

EXAMPLES::

sage: B = BraidGroup(5)

sage: B.epimorphisms(SymmetricGroup(5))

[Generic morphism:

From: Braid group on 5 strands

To: Symmetric group of order 5! as a permutation group

Defn: s0 |--> (1,5)

s1 |--> (4,5)

s2 |--> (3,4)

s3 |--> (2,3)]

Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

enriqueartal · 2023-07-27T08:58:34Z

Now the method epimorphisms returns a list of generic morphisms. I guess it was the only choice when this method was created, but now group homomorphisms exist. I wonder if it would be better to change it. In that case some GAP functions would be accessible.

tscrim · 2023-07-28T06:39:58Z

It doesn't seem like the homspace knows how to make itself using the more specific class. I would put that off for another ticket.

tscrim

Just a few more details. A number of your changes are not really useful (e.g., reordering the import statements in some arbitrary order (yes, I know it is alphabetical, but that is still an arbitrary choice)) that could create trivial conflicts with other changes, but it's already done. However, the stylistic changes will not get a positive review from me.

src/sage/groups/artin.py

src/sage/groups/braid.py

tscrim · 2023-09-05T01:01:25Z

src/sage/groups/braid.py

+            sage: c1 == b.permutation()
+            True
+
+        From a permutation it is also possible to recover the permutation braid::


Suggested change

From a permutation it is also possible to recover the permutation braid::

It also applies the canonical section to a permutation::

The current version implies a certain uniqueness that is simply not true. Consider $s^3$ versus $s$ for a simple transposition $s$.

I put another sentence. I really want to keep the expression permutation braid (or positive permutation braid) since is a common concept in braid group theory. Please check it.

enriqueartal · 2023-09-05T11:09:48Z

I applied you comments. One should be reviewed again. Probably not now, by I think that in general epimorphisms should give group homomorphisms and not generic ones.

tscrim

Thank you. I am going to approve this PR; just make the one last change and then I will set this to a positive review.

src/sage/groups/braid.py

Erase the period Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

tscrim · 2023-09-07T13:12:32Z

Thank you.

github-actions · 2023-09-07T13:49:52Z

Documentation preview for this PR (built with commit 29ca7a7; changes) is ready! 🎉

enriqueartal added 24 commits April 12, 2023 15:39

Add documentation

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Change _standard_lift_Tietze for braids

Define the braid of a either a permutation or an element in the symme…

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…tric group

More examples

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test error

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Merge branch 'sagemath:develop' into artin

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New methods for braids

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Erratas

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Merge pull request #3 from mkoeppe/gcc_13

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Gcc 13

Merge pull request #4 from mkoeppe/libgd_reduce_deps

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Libgd reduce deps

Allow gfortran13

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merge 10.1beta0

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some typos

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enriqueartal requested review from tscrim and miguelmarco July 24, 2023 16:55

enriqueartal added t: enhancement c: group theory s: needs review labels Jul 24, 2023

tscrim reviewed Jul 26, 2023

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enriqueartal changed the title ~~More improvements for braid groups involving permuationsv, and a 2-generator presentation~~ More improvements for braid groups involving permuations, and a 2-generator presentation Jul 27, 2023

enriqueartal changed the title ~~More improvements for braid groups involving permuations, and a 2-generator presentation~~ More improvements for braid groups involving permutations, and a 2-generator presentation Jul 27, 2023

enriqueartal and others added 6 commits July 27, 2023 09:15

Update src/sage/groups/artin.py

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Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

Update src/sage/groups/artin.py

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Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

Update src/sage/groups/braid.py

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Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

Update src/sage/groups/braid.py

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Co-authored-by: Travis Scrimshaw <clfrngrown@aol.com>

changes from the review

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use combinations from itertools

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enriqueartal added 3 commits July 31, 2023 16:24

Merge branch 'sagemath:develop' into artin

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enriqueartal mentioned this pull request Nov 25, 2023

More information on fundamental groups of plane curves enriqueartal/sage#7

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More improvements for braid groups involving permutations, and a 2-generator presentation #35985

More improvements for braid groups involving permutations, and a 2-generator presentation #35985

enriqueartal commented Jul 24, 2023

tscrim Jul 26, 2023

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enriqueartal commented Jul 27, 2023

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tscrim commented Sep 7, 2023

github-actions bot commented Sep 7, 2023

	An element of the Coxeter group or an element of a symmetric group.
	An element of the Coxeter group ``W``.

		The image of ``self`` under the natural projection map to ``W``
		as either a standard permutation or an element of a symmetric group.

		- ``isomorphism`` -- boolean (default ``False``). If ``True`` an isomorphism
		from ``self`` and the isomorphic group and its inverse are provided

	From a permutation it is also possible to recover the permutation braid::
	It also applies the canonical section to a permutation::

More improvements for braid groups involving permutations, and a 2-generator presentation #35985

More improvements for braid groups involving permutations, and a 2-generator presentation #35985

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enriqueartal commented Jul 24, 2023

📝 Checklist

⌛ Dependencies

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enriqueartal commented Jul 27, 2023

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