Dear Vít,
On Sun, Oct 06, 2013 at 11:03:38AM -0700, Vít Tuček wrote:
> I have implemented an algorithm for computing the minimal representatives
> of cosets W/W_S where W is the Weyl group of a Lie algebra and W_S is the
> Weyl group of its standard Levi subalgebra corresponding to the set of
> simple roots S. These posets have their place well established in the
> literature, e.g. they govern nilpotent Lie algebra cohomology of
> finite-dimensional modules and homomorphisms of (generalized) Verma
> modules. I use my routines for producing illustrations in my work and for
> checking theorems in low dimensions.
> I would like this code to be included into sage but I am a bit unsure
> about the API. Right now I have implemented methods of WeylGroup
> class parabolic_bruhat_graph(self, Levi_indices)
> and parabolic_poset(self, Levi_indices) which are basically just wrappers
> over _minimal_representatives(self,Levi_indices).
> Sometimes it is convenient (and is quite common in the literature) to
> specify the Levi part by the simple roots which are NOT contained in its
> root system. Should this be implemented in the API as well?
> On a related note, each standard parabolic subalgebra determines a
> (symmetric) grading on the Lie algebra (g = g_{-k} \oplus \cdots g_l) and
> any such grading determines a parabolic subalgebra. I have implemented
> method for RootSpace which splits the positive roots into the graded
> components, i.e. the result is a dictionary {j : g_j } Could this be
> included as well?
Thanks for offering to share your code!
Just some quick remarks to get started: the minimal representatives
themselves are already available for any Coxeter group as
"coset_representatives":
sage: W = SymmetricGroup(4)
sage: WI = W.coset_representatives([1,2], side="right")
sage: WI.cardinality()
4
Is the first part of your contribution about those, or about building
the bruhat order on those coset representatives? In the later case,
are you using "coset_representatives" to get the elements themselves?
For the user interface, a natural choice could be to add the feature
to the existing bruhat_poset method of FiniteCoxeterGroups:
sage: W.bruhat_poset(index_set=[1,2,4])
This would be consistent with the syntax for the quantum_bruhat_graph
method of WeylGroups. Here, I am assuming your algorithm work for any
(finite) Coxeter group, right?
Further comments anyone?
For the positive roots: the system already knows that there is a grading:
sage: W = CoxeterGroups().example()
sage: R = RootSystem(["A",3]).root_lattice()
sage: R.positive_roots()
<sage.combinat.backtrack.TransitiveIdealGraded instance at 0x7992440>
What's really missing is to upgrade TransitiveIdealGraded so that one
can easily ask for all elements of a given depth. See also the related
ticket http://trac.sagemath.org/ticket/6637.
> In some areas of mathematics it is customary to actually draw these posets
> with nodes replaced by labelled Dynkin diagrams with nodes not
> corresponding to the Levi part crossed. See
> e.g. http://arxiv.org/abs/1303.1307
> I haven't looked into graph drawing of sage nor at Dynkin diagrams, but in
> case this feature also has a chance to be included in sage I am willing to
> try to implement it.
It should not be too hard indeed!
> I'll post patches as soon as I get properly acquainted with sage
> patch & build system.
Which is in a state of flux, since we are currently switching
development workflow (including a switch from mercurial to git).
See: http://trac.sagemath.org/ticket/13015
Cheers,
Nicolas
--
Nicolas M. Thiéry "Isil" <nthi...@users.sf.net>
http://Nicolas.Thiery.name/
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