Jucys-Murphy Elements for Brauer Algebra

[ghseeli]

Nazarov's 1996 paper titled "Young's Orthogonal Form for Brauer's Centralizer Algebra," outlines a form for the Jucys-Murphy elements that are analogous to the JM elements for the symmetric group algebra. Implementing these into Sage would require a few lines of code but would prove very useful for work done with the Brauer algebra.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​e138144	Added documentation and a doctest for coercion
​5b83e6f	Merge branch 't/18762/create_coercion_between_diagram_algebras_and_the_symmetric_group_algebra' into t/18798/jucys_murphy_elements_for_brauer_algebra
​d01322b	Added necessary doctests to many more __ methods
​a4fe9c2	fixed a few typos
​e527bbe	Added attribution to authors in source
​43252e7	Merge branch 't/18720/change_diagram_algebra_basis_set_partitions_from_list_to_generator' into t/18762/create_coercion_between_diagram_algebras_and_the_symmetric_group_algebra
​26abcdf	Merge branch 't/18762/create_coercion_between_diagram_algebras_and_the_symmetric_group_algebra' into t/18798/jucys_murphy_elements_for_brauer_algebra
​639c9f6	fixed an odd typo

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​d4d3a09	accidentally deleted crucial statement for coercion. I have reinserted it.
​7160ffe	fixed merge conflicts
​2adcfef	oops! Didn't completely merge
​781c643	Merge branch 'develop' into t/18762/create_coercion_between_diagram_algebras_and_the_symmetric_group_algebra
​888f1a2	Merge branch 'develop' into t/18762/create_coercion_between_diagram_algebras_and_the_symmetric_group_algebra
​6e7cb21	Created conversions of SymmetricGroupAlgebra to DiagramAlgebras as well as appropriate coercions where it makes sense
​ea857ff	added more negative unit tests for conversion from symmetric group algebra to diagram algebra. One test does not pass
​a4c199d	added the necessary extra colon after tests
​14f80d3	Reviewer changes and fixing coercions.
​ab6b680	Merge branch 't/18762/create_coercion_between_diagram_algebras_and_the_symmetric_group_algebra' into t/18798/jucys_murphy_elements_for_brauer_algebra

[ghseeli]

Sorry, this git history is hard to read! The only change this ticket should include over 18762 is the jucys_murphy method within the BrauerAlgebra class.

[tscrim]

Is this ready for review? A few things from a quick lookover: Make the reference key [N] longer to avoid potential conflicts and docstrings should be declarative (so it will start with Return).

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​b17c1c9	Updated the Nazarov reference to be abbreviated as Naz instead of N.
​15aa5a1	Made docstring declarative.

[ghseeli]

Yeah, this is ready for review. Thank you for the suggestions thus far!

[tscrim]

A couple of suggestions:

     def jucys_murphy(self, j):
         r"""
-        Return a generalized Jucys-Murphy elements for the Brauer algebra.
-        These are outlined in [Naz]_.
+        Return the ``j``-th generalized Jucys-Murphy element of ``self``.
 
+        Could you give a more detailed description of how the elements are defined?
+
         REFERENCES:
 
         .. [Naz] Maxim Nazarov, Young's Orthogonal Form for Brauer's Centralizer
-               Algebra. Journal of Algebra 182 (1996), 664--693.
+           Algebra. Journal of Algebra 182 (1996), 664--693.
 
         EXAMPLES:
 
             sage: z = var('z')
             sage: B = BrauerAlgebra(3,z)
             sage: B.jucys_murphy(1)
             1/2*z - 1/2
             sage: B.jucys_murphy(3)
-            -B{{-3, -2}, {-1, 1}, {2, 3}} - B{{-3, -1}, {-2, 2}, {1, 3}} + B{{-3, 1}, {-2, 2}, {-1, 3}} + B{{-3, 2}, {-2, 3}, {-1, 1}} + (1/2*z-1/2)*B{{-3, 3}, {-2, 2}, {-1, 1}}
+            -B{{-3, -2}, {-1, 1}, {2, 3}} - B{{-3, -1}, {-2, 2}, {1, 3}}
+            + B{{-3, 1}, {-2, 2}, {-1, 3}} + B{{-3, 2}, {-2, 3}, {-1, 1}}
+            + (1/2*z-1/2)*B{{-3, 3}, {-2, 2}, {-1, 1}}
         """
-        B = self
-        return (B._q-1)/2 + sum(B([[i,-j],[j,-i]]) - B([[i,j],[-i,-j]]) for i in range(1,j))
+        I = self._indices
+        one = self.base_ring().one()
+        return ((self._q-1)/2
+                + self._from_dict({I([[i,-j],[j,-i]]): one for i in range(1,j)}, remove_zeros=False)
+                - self._from_dict({I([[i,j],[-i,-j]]): one for i in range(1,j)}, remove_zeros=False))

Do you think we should also make this a cached method? What about when j > n, what happens then (or is it even defined)?

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​5406172

Added an assert statement and took into account the one element of the base ring.

[ghseeli]

Alright, I added a small explanation to the docstring, and also took into account the one element of base ring. However, the problem with your _from_dict rewrite is that it does not pass the doctests. This is because when you call the element constructor, it "fills in" information you don't give it, but the BrauerDiagram class does not do this. Is there a reason to prefer using the _from_dict approach? Finally, I believe it probably could be a cached method, although it also isn't a terribly intense computation.

[tscrim]

I made some minor tweaks to the documentation. The reason why I use the _from_dict to avoid the coercion framework, and it has a two-fold advantage of being faster and not invoking the coercion framework (which has in the past led to annoying consequences). My changes also fix a bug that the first Jucys-Murphy element was not in the Brauer algebra. If you're happy with my changes, then you can set a positive review.

---

New commits:

​a660f53	Merge branch 'u/ghseeli/jucys_murphy_elements_for_brauer_algebra' of trac.sagemath.org:sage into u/ghseeli/jucys_murphy_elements_for_brauer_algebra
​9613ee3	Tweaks to the JM code and fixing a doctest.

[ghseeli]

Alright. I am satisfied with these changes. Good catch on the first element!