Root systems improvements

[nthiery]

Patch taken from Sage-Combinat: ​http://combinat.sagemath.org/patches/file/tip/root_systems-4326-nt.patch

Depends on: #6136 #6253 #6250 #5891

---

Documention:

• quickref + links in sage.combinat.root_system
• Long introduction in CartanTypes?
• ...

Cartan Types:

• Object oriented clean up: each cartan type has its own class (in .type_....py) which contains all its specific data (dynkin diagram, ascii art, ...). All the dispatch logic is now concentrated in the CartanType? factory.
• fixed the definition of rank for affine types (Anne Schilling)
• systematic implementation of the classical type underlying an affine type (Anne Schilling)
• New methods: is_untwisted_affine, special_node, a, acheck, translation_factors, symmetrizer, row_annihilator col_annihilator (partly Nicolas Borie)
• Relabelled Cartan types (with composition, classical, special_node, dual)
• Use A~... B~* BC~ convention for affine types; Kac' convention implemented by renaming them (see CartanType??)
• F3 is nonexistent so use F4 in one test (Dan Bump)
• ascii art for reducible (Dan Bump), relabelled, and dual Cartan types

Root systems:

• Preliminary plots (Nicolas Borie)
• New methods for affine root systems (mostly Nicolas Borie): null_(co)root, level
• RootSystem?(["A",3,1]) returns None rather than the ambient space for type A_3 (which was wrong!)
• positive and negative roots for all (finite) root lattice realizations

Coxeter groups:

• New categories: (Finite) CoxeterGroups?, (Finite, Affine) WeylGroups? standardized methods: first_descent, has_descent, descents, reduced_word, length, from_reduced_word, with systematic associated test (test_has_descent, ...) simple_reflections, simple_projections, coset_representatives, binary_factorisations, ... (many of them were extracted and generalized from WeylGroup?)
• lower and upper cover for Bruhat order (Steve Pon)
• affine stanley symmetric functions for types A, A affine
• Documentation (with help from Qiang Wang, Nicolas Borie)

The following are not yet addressed, and will be bumped to a subsequent patch:

DynkinDiagram?:

• allow for slicing notation for column/row extraction: c[i,:]

AmbientSpace?:

• fundamental coweights by appropriate scaling of the fundamental weights
• embedding coweight lattice

WeightLatticeRealization?

• scalar product with coweight lattice in finite dimension

Classical case:

• reverse map to coroot space and coroot lattice by scalar product with the fundamental weights
• associated coroot in the root and weight space
• s_\alpha on the (co)root and (co)weight lattice for any root \alpha
• highest_coroot returning self.coroot_lattice().highest_root()

Affine case:

• affine ambient space
• Fix:

    sage: R = RootSystem(["A",2,1]).weight_lattice()
    sage: R.rho_classical()
    ...
    TypeError: unsupported operand parent(s) for '*': 'Rational Field' and 'Weight lattice of the Root system of type ['A', 2, 1]'
  

Categorification of RootLatticeRealization? / ... New category CoxeterGroupModules? Support for non crystalographic root systems

[bump]

Normally I would have waited for the category patches to be merged before reviewing this patch. However I recieved an email from Tom Boothby urging me to do the review now, so here it is.

This review is based on the version of the patches in the combinat queue. This is because it depends on patches that have not been merged yet.

After qpushing the combinat queue up to this patch but not beyond, all tests pass. This is with Sage 4.1 and the last changeset is this one:

changeset:   1520:188022ff52b9
tag:         tip
user:        Nicolas M. Thiery <nthiery@users.sf.net>
date:        Tue Jul 21 01:13:42 2009 +0200
summary:     Update

The patch adds quite a bit of new functionality for working with Coxeter groups and affine Weyl groups. The following new files are added. There are new categories added for CoxeterGroups? and WeylGroups?. There is an extensive ChangeLog? in the comments at the beginning of the patch.

Since the patch is over 11,000 lines of code, there could very well be bugs in it. However it probably does not introduce significant new bugs in the portion of the code that deals with classical root systems, since I used it extensively during the spring of 2009 in connection with #5794. Every classical Cartan type and many reducible types have been created and worked with. What problems I found then were fixed. Moreover the portion of the code that deals with affine root systems was similarly very tested by Anne Schilling in connection with affine crystals.

Therefore the most uncertainty in my is with the new functionality for Coxeter groups. I will mention one "wish" in this direction, which is that in addition to implementing the Bruhat covers, the Bruhat order be properly implemented. This could be done efficiently using Proposition 1.1 in Stembridge, A Short Derivation of the Möbius Function for the Bruhat Order, Journal of Algebraic Combinatorics 25, (2007).

This wish is not a reason to hold up merging the patch. Rather the patch should be merged as soon as possible and such changes can be made later.

[mvngu]

Let me get this straight. Tickets #6136, #6253, #6250, and #5891 must first be merged before merging this ticket?

[nthiery]

Replying to mvngu:

Let me get this straight. Tickets #6136, #6253, #6250, and #5891 must first be merged before merging this ticket?

Indeed. Should we use a specific subject for patches that have a positive review, but have dependencies on not yet merged tickets? Something like:

[with patch, positive review, depends on #6136, #6253, #6250, #5891] ...

[mvngu]

Replying to nthiery:

Indeed. Should we use a specific subject for patches that have a positive review, but have dependencies on not yet merged tickets? Something like:

[with patch, positive review, depends on #6136, #6253, #6250, #5891] ...

No, not really. I just want to double check since the patch is rather huge and I was concerned about it getting bit rotten. Anyway, people who uses the merge script would not be able to easily tell whether a positive-reviewed ticket has other dependencies.

[nthiery]

Latest version of the patch from the Sage-Combinat patch server (no change since Dan's review)

[nthiery]

Fix ClassicalWeylSubgroup? and remove unneeded long time, as spotted by Mike

[mhansen]

The new patch looks okay to me.