#19506: Implement cellular algebras
-------------------------------------+-------------------------------------
Reporter: tscrim | Owner: sage-combinat
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.10
Component: categories | Resolution:
Keywords: cellular algebra | Merged in:
Authors: Travis Scrimshaw | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
public/categories/cellular_algebras-19506|
cbdbc2b85f1954c17701280d59d70def65d1669c
Dependencies: | Stopgaps:
-------------------------------------+-------------------------------------
Comment (by tscrim):
I was originally following the lecture notes by Xi above, where RSK was
define the basis (see example 5). However my test suite actually told me
this was incorrect. So instead I switched to the seminormal basis, which
as far as I can tell, is the specialization of the Murphy basis of the
Hecke algebra.
To implement an object in cellular algebras, one needs to implement the
following:
- `cell_poset` which returns the poset parameterizing the cells.
- `cell` which takes an element `la` in the cell poset and returns the
cell `M(la)`.
- One of the following:
* `_to_cellular_element` which takes a basis index `i` and return an
element in `cellular_basis`.
* `_from_cellular_index` which takes `(la, s, t)` and returns an element
of the algebra.
* `cellular_basis` which returns an algebra with a basis indexed by
`(la, s, t)` and has coercions to and from the algebra.
The first two are simply data, but the non-trivial part is the third one.
However my current framework in a way assumes a distinguished cellular
basis, but that is not to say it limits the algebra to one cellular basis.
Actually, now that I look over it again, my mechanism for
`to_cellular_element` is broken and I will fix that shortly.
The next step would be to implement the representations, bilinear form,
decomposition matrix, and Cartan matrix.
--
Ticket URL: <http://trac.sagemath.org/ticket/19506#comment:5>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
--
You received this message because you are subscribed to the Google Groups
"sage-trac" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/sage-trac.
For more options, visit https://groups.google.com/d/optout.
