#15272: Bruhat posets and Bruhat graphs for parabolic subgroups of finite Weyl
groups
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Reporter: vittucek | Owner:
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.2
Component: combinatorics | Resolution:
Keywords: | Merged in:
Authors: Vít Tuček | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/vittucek/ticket/15272 | 472c0908fba197fe9bb73742caab1ebfcde27b53
Dependencies: | Stopgaps:
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Changes (by vittucek):
* cc: nthiery, bump (added)
* commit: => 472c0908fba197fe9bb73742caab1ebfcde27b53
Old description:
> This patch extends the functionality of bruhat_poset and introduces new
> parabolic_bruhat_graph for finite Weyl groups.
>
> Let W be a finite Weyl group and let W_S be the subgroup of W generated
> by reflections associated with a subset S of simple roots. Then the
> cosets W / W_S have unique representatives of minimal length which are
> ordered by the Bruhat order of W. Similarly for W_S \ W. These poset
> structures appear in many places, e.g. intersection cohomology of
> generalized flag varieties or nilpotent Lie algebra cohomology.
>
> This patch adds a parameters index_set (= S) and side (left / right).
>
> ----
>
> Introducing parabolic_bruhat_graph is ugly. Ideally, one would just
> extend the existing bruhat_graph. However, this method is based upon
> bruhat_interval which belongs to categories/coxeter_groups.py
>
> I was unsure where to put the code, which I haven't written yet nor which
> I need in the foreseeable future anyway. Since it seems that the best
> course of action would be to implement class (or category?) for parabolic
> subroot systems / groups I think that one more method for Weyl group is
> not much of an issue.
New description:
This patch adds method minimal_representatives and extends the
functionality of bruhat_poset and bruhat_graphs.
Let W be a finite Weyl group and let W_S be the subgroup of W generated by
reflections associated with a subset S of simple roots. Then the cosets W
/ W_S have unique representatives of minimal length which are ordered by
the Bruhat order of W. Similarly for W_S \ W. These poset structures
appear in many places, e.g. intersection cohomology of generalized flag
varieties or nilpotent Lie algebra cohomology.
This patch adds a parameters index_set (= S), crossed_nodes and side (left
/ right).
--
Comment:
New commits:
||[http://git.sagemath.org/sage.git/commit/?id=7bb7c3b3c9864b6a82b9bc87464d9076ad05ebb1
7bb7c3b]||{{{Decorate ClassicalWeylSubgroup.simple_reflections() with
@cached_method.}}}||
||[http://git.sagemath.org/sage.git/commit/?id=6db7b6d186ef715f647c848820b22d4cc7c23110
6db7b6d]||{{{First version of minimal coset representatives.}}}||
||[http://git.sagemath.org/sage.git/commit/?id=472c0908fba197fe9bb73742caab1ebfcde27b53
472c090]||{{{Add parabolic minimal_coset_representatives, bruhat_graph and
bruhat_poset}}}||
--
Ticket URL: <http://trac.sagemath.org/ticket/15272#comment:4>
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