#10305: Add rings for the center of the symmetric group algebras
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Reporter: mguaypaq | Owner: mguaypaq
Type: enhancement | Status: new
Priority: minor | Milestone: sage-4.6.1
Component: combinatorics | Keywords: combinatorics, rings,
symmetric functions
Author: Mathieu Guay-Paquet | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Here is some preliminary code to implement the center of the symmetric
group algebras, various bases for this and related rings, and coercions
between these and to/from the ring of symmetric functions.
Things that I plan to do:
* Fill in all of the TODO doctests with actual doctests.
* Figure out why some of the coercions aren't discovered by the coercion
framework. (See the last few examples at the top of the file.)
* `SymmetricGroupAlgebraCenter` does not actually have a multiplicative
identity as currently implemented, so it should probably by in a category
other than `GradedAlgebrasWithBasis`.
* Reimplement `ConjugacyClassBasis` and `OrthogonalIdempotentBasis` for a
fixed symmetric group over the integers. Then, the conversion table can be
cached, the elements can be represented by lists of coefficients (possibly
numpy arrays) instead of dictionaries, and faster linear algebra over the
integers can be used. This results in massive (several orders of
magnitude) speedups.
* Possibly add the option of saving some of the conversion tables to
disk, as they can be large and expensive to compute.
* Add the relevant classes to `sage/combinat/all.py`.
Note that this depends on #7980 and #10304.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10305>
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