Dear developers,
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?
Last thing:
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.
I'll post patches as soon as I get properly acquainted with sage patch &
build system.
Best regards,
Vít Tuček
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