Implementation of the Lenart--Postnikov alcove path crystal

[sagetrac-brant-c-jones]

This is an implementation of the Lenart--Postnikov alcove path model as described in:

A combinatorial model for crystals of Kac-Moody algebras. Trans. Amer. Math. Soc. 360 (2008).

It also implements to_coroot_lattice_morphism() and associated_coroot() in root_lattice_realization.py.

Depends on #8911 (trac_8911_categorification_crystals-as.patch).

CC: @sagetrac-sage-combinat brant@math.ucdavis.edu @rlmill

Component: combinatorics

Keywords: combinat, crystals

Author: Brant Jones

Reviewer: Anne Schilling

Merged: sage-4.5.2.alpha0

Issue created by migration from https://trac.sagemath.org/ticket/8984

[sagetrac-brant-c-jones]

Description changed:

--- 
+++ 
@@ -4,3 +4,5 @@
 
 It also implements to_coroot_lattice_morphism() and associated_coroot() in root_lattice_realization.py.
 
+Depends on #8911 (trac_8911_categorification_crystals-as.patch).
+

[anneschilling]

comment:3

Test whether Brant gets this message.

[anneschilling]

Changed keywords from none to combinat, crystals

[anneschilling]

comment:5

Thank you for implementing the alcove path model by Lenart and Postnikov.
This will be a useful addition to sage.

It might be useful to add a few more words about the model you implemented
in the documentation of ClassicalCrystalOfAlcovePaths. For example, you could
add that these are highest weight crystals for classical types
A_n, B_n, C_n, D_n and the exceptional types F_4, G_2, E_6,
E_7, E_8.

Also, for the user it would be helpful to say how precisely one should
enter the input data. For example, you could say

INPUT:

• cartan_type is the Cartan type of a classical Dynkin diagram
• highest_weight is a dominant weight as a list of coefficients of
  the fundamental weights Lambda_i

It might also be good to briefly describe how the crystal elements
are represented so that the user can interpret the output.

Some technical comments:

(1) In combinat/crystals/alcove_path.py, it might be safer to only
import the methods/classes that you really need for:

from sage.rings.integer import * (which appears twice, so please remove one!)
from sage.misc.misc import *
from sage.calculus.calculus import *

(2) All methods need EXAMPLES or TESTS. Please add them to the following
methods in combinat/crystals/alcove_path.py for

classcall
get_initial_chain
fold
compare_graphs

(3) Perhaps remove the commented out lines by # in

init
list

(4) Add extra line after EXAMPLES:: get_chain_from_subset

(5) Remove the commented out weight function

(6) You need TESTS or EXAMPLES to_coroot_lattice_morphism in
sage/combinat/root_system/root_lattice_realization.py

[sagetrac-brant-c-jones]

comment:6

Attachment: trac_8984_crystals_alcove_path_model_bj.patch.gz

I have implemented all of the suggestions given by the reviewer above.

Please review the new version.

[anneschilling]

Reviewer: Anne Schilling