Implementation of the Lenart--Postnikov alcove path crystal

[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).

[aschilling]

Test whether Brant gets this message.

[aschilling]

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

[brant.c.jones]

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

Please review the new version.

[aschilling]

This patch implements the Lenart-Postnikov model for highest weight crystals for finite-dimensional Lie algebras. There are extensive tests to test that this gives the same crystal graph as other models.

All tests pass with sage-4.4.2 and the sage-combinat queue applied to this patch.

[mhansen]

I had to backout this change from 4.4.4 for now. I was getting weird failures with random_element in matrix_group.py. I'm trying to figure out why this patch was causing it.

[aschilling]

Replying to mhansen:

Hi Mike,

What is the status on this now? Do you know why there were the strange failures in random_element in matrix_group.py?

Anne

[drkirkby]

See #9310

[aschilling]

Replying to drkirkby:

See #9310

Is this patch the cause of the failure in #9310 though? It was not merged into sage-4.4.4 and still the error is there.

[nthiery]

Hi Robert!

Any chances to merge this patch, since it does not seem any more related to the failures than any other?

Thanks!

[rlm]

sage-4.5 is in feature freeze mode. Nothing but essential fixes will be merged until final release. I am strongly suggesting that the next release be patches to the sage library only (other than essential spkg fixes), and tickets like these deserve to go in then. (I would have had an alpha for such tickets in the 4.5 series, but the spkg issues are already holding things up long enough.)