#14261: Iwahori-Hecke algebra with several bases
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Reporter: brant | Owner: sage-combinat
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.12
Component: combinatorics | Resolution:
Keywords: Iwahori Hecke | Merged in:
algebra | Reviewers: Andrew Mathas, Brant
Authors: Brant Jones, | Jones, Travis Scrimshaw
Travis Scrimshaw | Work issues:
Report Upstream: N/A | Commit:
Branch: | Stopgaps:
Dependencies: #13735 #14014 |
#14678 #14516 |
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Comment (by nthiery):
Just a tiny bit of food for thoughts: In symmetric functions, we are
facing a similar issue when it comes to implementing the plethysm: one
needs to know which parameters in the ground field are of rank 1 (in
which case p_k(x)=x^k) or of rank 0 (in which case p_k(x)=x). And of
course specializing a rank 1 parameter to a complex number does not
play well with plethysm.
The approach we took in MuPAD-Combinat for this particular issue (and,
I think, should be implemented in Sage) is to have the user specify
explicitly what the plethysm does on the ground field, and this worked
pretty well.
In practice the user had to provide a glorified field with a plethysm
method. This was typically "ExpressionField" (roughly the equivalent
of Sage's symbolic ring) glorified as the facade parent
"ExpressionFieldWithDegreeOneElement(variables)" that added a
"plethysm" method which did an appropriate substitution on the
variables. See e.g. p. 4 of [1].
So one could imagine a similar approach where: if the user wants the
full featured Hecke algebra, (s)he would have to provide explicitly a
method implementing the bar involution on the ground field (assuming
of course it actually makes sense!). This would give the flexibility
to handle several cases: free parameters without
doing nonsense like inverting other variables in the ground field,
parameters on the unit
circle (taking complex conjugation as bar involution), ...
Of course, a default implementation of the bar involution could be
automatically provided in the easy cases. One point to discuss would
be the interface: do we want to create a glorified field with a bar
method, or just provide the bar method separately.
Again, just some food for thoughts; I am not saying it's necessarily
the right approach for the problem at hand.
Cheers,
Nicolas
[1] http://mupad-combinat.sourceforge.net/Papers/TutorialSymFcts.pdf
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Ticket URL: <http://trac.sagemath.org/ticket/14261#comment:39>
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