Kazhdan-Lusztig polynomials, Bruhat order, and related features

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This patch includes algorithms for the Bruhat order, Kazhdan-Lusztig polynomials, improvements to the __repr__ method of WeylGroup? elements, and other enhancements.

Mike Hansen is working on an interface to coxeter3, which is be able to compute Kazhdan-Lusztig polynomials rather faster. However I think this patch still contains things that are needed.

For discussion see this thread:

 http://groups.google.com/group/sage-combinat-devel/browse_thread/thread/d324f6fcb6d2a436?hl=en

This patch depends on #7753 and #7754.

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I made a minor change so that the Kazhdan-Lusztig computation doesn't hang in the affine case. I've only done much testing for finite Weyl groups but I suppose it is correct in the affine case.

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I have revised the patch. It now depends on #7753 and #7754. The revised patch makes use of the Bruhat order as implemented in #7753 and makes the Kazhdan-Lusztig polynomials using @cached_method. Other changes allow the base ring to be a LaurentPolynomialRing.

The patch is much faster now, something like 50% faster.

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Rebased to Sage 4.3.1.

[roed]

Looks good. Here are a few comments. After these are addressed, I'll be happy to give this a positive review.

• sage/combinat/kazhdan_lusztig.py
  • typo in your e-mail address.
  • the method of determining KL._base_ring_type seems a little strange. Why not use is_Polynomial and isinstance(q, LaurentPolynomial?)?
  • KazhdanLusztigPolynomial? should inherit from SageObject?. That allows pickling, etc.

• sage/combinat/root_system/weyl_group.py *In WeylGroup_gens, __classcall_ needs another trailing underscore. Include a doctest to make sure that this feature works!
  • Can you include a doctest in WeylGroupElement.__repr__? I know it's tested elsewhere, but...

In general, do you have a reason to use __call__ explicitly, rather than parentheses? Similarly, you don't need to explicitly call repr: using %s in a string will do that for you automatically.

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I addressed most of David Roe's comments.

But the email address is not a typo.

Brant Jones is also looking at the patch.

[brant.c.jones]

Patch review: trac_7751

This patch implements Kazhdan--Lusztig polynomials and R-polynomials associated to pairs of Weyl group elements in arbitrary type using standard recursive formulas. I have verified the results of this code against the Kazhdan--Lusztig polynomials produced by GAP/Chevie for all pairs of elements in finite type A_4. I also verified this code against GAP/Chevie for all pairs in affine type A_2 (having 3 generators) for all pairs of elements with Coxeter length less than or equal to 5. This patch correctly implements useful mathematics and the documentation includes references to relevant mathematical literature.

-- Brant Jones

[roed]

I agree. Positive review.

[mpatel]

Applying the patch to  this hierarchy (minus #8232), I get

patching file sage/combinat/root_system/weyl_group.py
Hunk #5 FAILED at 145
1 out of 13 hunks FAILED -- saving rejects to file sage/combinat/root_system/weyl_group.py.rej

The reject:

--- weyl_group.py
+++ weyl_group.py
@@ -138,6 +146,7 @@
         self.n = lattice.dimension() # Really needed?
         # MatrixGroup_gens takes plain matrices as input. So we can't do:
         #MatrixGroup_gens.__init__(self, list(self.simple_reflections()))
+        self._prefix = prefix
         MatrixGroup_gens.__init__(self, [self.morphism_matrix(self.lattice().si
mple_reflection(i)) for i in self.index_set()])
 
     @cached_method

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If this conflict occurs, you may resolve just making sure the line self._prefix = prefix occurs somewhere in the init method of WeylGroup?.

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Correction: the line self._prefix = prefix should be somewhere in the init method of WeylGroup_gens (not WeylGroup).

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Kazhdan-Lusztig polynomials, Bruhat order, improved __repr__ for WeylGroups?.

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Patch rebased to sage-4.3.3.alpha0. This fixes the conflict reported by mpatel with no other changes.

[mvngu]

Daniel, I have committed the attachment  kazhdan_lusztig.patch in your name, since the patch doesn't contain your full name.