#8442: Lie Methods and Related Combinatorics (tutorial)
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Reporter: bump | Owner: bump
Type: enhancement | Status: needs_info
Priority: major | Milestone: sage-4.6.1
Component: documentation | Keywords:
Author: Daniel Bump | Upstream: N/A
Reviewer: Minh Van Nguyen, Mark Jordan | Merged:
Work_issues: |
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Comment(by bump):
> One is about a way to connecting Lie functionality in GAP to the one in
Sage. Anything on this?
One issue with GAP on Sage is that the interface needs a lot of work.
There is a lot of power in GAP that can't be accessed from Sage because of
this.
The Lie theory in Sage is mostly written from scratch. But here is an
example where GAP is
involved in the background. We have a class for WeylGroups in
weyl_group.py. This
class inherits from MatrixGroup_gens which in turn inherits from
MatrixGroup_gap. So
GAP is involved in Weyl Groups.
> It would also be good if anything is said regarding the optional Sage
package lie (by Marc van Leeuween). Is it right that basically anything
doable in lie can be done in Sage? In particular, lie can compute
decompositions of, say, a tensor product of two representations into
irreducibles. It's not clear to me whether one can do this in Sage
(without lie).
I am not sure whether everything that is doable with LiE is doable with
Sage but I do think that anything that is needed from LiE is either in
Sage already or (if needed) should be reimplemented. What is in Sage is a
pretty complete toolkit for finite-dimensional representations of Lie
groups. Decomposing a tensor product into irreducibles is just the
multiplication in the WeylCharacterRing. This is addressed in the
tutorial. See:
http://match.stanford.edu/bump/thematic_tutorials-
js/lie/weyl_character_ring.html#tensor-products-of-representations
Have a look also at the branching rules.
http://match.stanford.edu/bump/thematic_tutorials-
js/lie/branching_rules.html
LiE has some functionality for working with Kazhdan-Lusztig polynomials,
but that is in Sage, as fast as LiE (though not as fast as Coxeter3). LiE
has alternate methods of computing Weyl Characters including use of
Demazure characters. Some version of the Demazure character is in the
crystal code, but it would also be easy and perhaps useful to add a method
to the WeightRing. But it is not urgently needed. Sage uses the
Freudenthal multiplicity formula to compute the character.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8442#comment:45>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
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