Implement path realization for affine crystals

[tscrim]

Depends on #14402

CC: @sagetrac-sage-combinat @bsalisbury1 @anneschilling

Component: combinatorics

Keywords: affine crystals, days49

Author: Travis Scrimshaw

Reviewer: Ben Salisbury, Anne Schilling

Merged: sage-5.12.beta1

Issue created by migration from https://trac.sagemath.org/ticket/14573

[anneschilling]

comment:2

Hi Travis,

The name of the class is not good. I would rather name such a class KyotoPathModel. Also, there are several things stated wrong in the documentation. In addition, several methods do not explain their input. And why only input one B crystal? This model works in much greater generality.

It might be good if you could talk to people first to see what the needs are and what the best way to implement them are.

Best wishes,

Anne

[tscrim]

Changed dependencies from #14402 #14516 to #14402

[tscrim]

comment:4

Hey Anne,

New version of the patch which does not depend on #14516 and addresses what we discussed.

Best,

Travis

[tscrim]

comment:5

New version with changes from Ben.

[bsalisbury1]

Changed keywords from affine crystals to affine crystals, days49

[bsalisbury1]

Reviewer: Ben Salisbury

[anneschilling]

Changed reviewer from Ben Salisbury to Ben Salisbury, Anne Schilling

[anneschilling]

comment:8

Hi Travis,

I have some further comments:

• The commit message is the merge of two patches; you might want to edit it!

• There is a typo "canonial" -> canonical

• Please explain what you mean by the bar on the weights. Also it would be a good idea to mention that \Lambda_i are the fundamental weights.

• Please mention that you are using the anti-Kashiwara convention for tensor factors (since I think your documentation differs in this from some of the literature and this might be confusing to users).

• Perhaps use capital N also in the equation before the sentence "We note by Lemma 10.6.2 in [HK02] ....".

• The documentation says B is a list of perfect crystals. But as far as I can see, in the examples the input is a single crystal. Could you please clarify? As I told Travis at some point in person, I think it would be a good idea to allow a list of crystals of the same level. If [B1,B2,....,Bk] would be such a list, one would use \cdots \otimes B_1 \otimes B_2 \otimes \cdots B_k \otimes B_1 \otimes \cdots B_k \otimes B(\lambda) as the model for the affine highest weight crystal.

• Could you add the affine grading? This would be useful since it is equal to the energy function!

Thank you,

Anne

[tscrim]

comment:9

All are done. I added an example showing it works with multiple crystals, and affine grading is inherited from the TensorProductOfCrystalsElement.

[anneschilling]

comment:10

Hi Travis and Ben,

I just discussed with Travis and he needs to change three more things:

• \mathfrankg needs to be fixed

• Please add the definition of \bar{\Lambda}

• For the input B can be a single crystal or a list of crystals.

Ben, Travis will make these changes tomorrow and once he has done it, you can set it to positive review.

Anne

[tscrim]

comment:11

Attachment: trac_14573-path_realizations-ts.patch.gz

Fixed and ready for a final review.

[jdemeyer]

Merged: sage-5.12.beta1