#15536: Implement symplectic and orthogonal bases of Sym
-------------------------------------+-------------------------------------
       Reporter:  tscrim             |        Owner:  sage-combinat
           Type:  enhancement        |       Status:  needs_review
       Priority:  major              |    Milestone:  sage-6.10
      Component:  combinatorics      |   Resolution:
       Keywords:  days54, sym        |    Merged in:
        Authors:  Travis Scrimshaw   |    Reviewers:
Report Upstream:  N/A                |  Work issues:
         Branch:                     |       Commit:
  u/tscrim/sp_orth_broken            |  cac1d88d687705fa90de0edac1b472f6e2d623b4
   Dependencies:  #17096             |     Stopgaps:
-------------------------------------+-------------------------------------
Changes (by tscrim):

 * cc: SimonKing, nthiery (added)
 * commit:  c968782e777fe17f42c5d316412e15feba1255cd =>
     cac1d88d687705fa90de0edac1b472f6e2d623b4
 * branch:  public/combinat/sf/sp_orth => u/tscrim/sp_orth_broken


Comment:

 Okay, I've done some reworking of things at the category level, including
 adding a `counit_by_coercion` for realizations.

 However I get MRO issues when dealing with the finite fields that I don't
 know how to deal with. I've convinced myself that the categories are
 almost correct; I want to make this change
 {{{#!diff
          cat = HopfAlgebras(self.base().base_ring())
          return [self.base().Realizations(),
                  cat.Commutative().WithBasis(),
 -                cat.Graded().Realizations()]
 +                cat.Commutative().Graded().Realizations()]
 }}}
 But if I do that, I get `KeyError: (225536, 70)` from the c3 controlled!

 So I don't know what to do at this point. Nicolas, Simon, do you have any
 ideas?
 ----
 New commits:
 
||[http://git.sagemath.org/sage.git/commit/?id=cac1d88d687705fa90de0edac1b472f6e2d623b4
 cac1d88]||{{{Doing some surgery on the categories.}}}||

--
Ticket URL: <http://trac.sagemath.org/ticket/15536#comment:35>
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