Implement tensor product of cystals categories and a better search for highest weight elements

[tscrim]

Currently in a tensor product of crystals, we search through all elements and filter. However if given a highest weight element in a tensor product a (x) b, then b must be highest weight. This means we can use a backtracing algorithm to find all highest weight elements which this ticket implements.

This first implements the categories of tensor products of crystals (currently all crystal subcategories are monoidal under tensor products).

This also changes the output of the current highest_weight_vectors() to return a tuple since we don't want to allow modifications to the output of a @cached_method.

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

Component: combinatorics

Keywords: crystals, tensor products

Author: Travis Scrimshaw

Branch: c9df470

Reviewer: Anne Schilling, Nicolas M. Thiéry

Issue created by migration from https://trac.sagemath.org/ticket/16451

[tscrim]

comment:1

As far as I can tell, this is independent of #16001.

Nicolas, can you make a quick check to make sure I'm doing everything correctly? If I am, I think I've found a bug for Modules:

sage: Modules(QQ).TensorProducts()
Category of objects

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New commits:

a68457f	Implemented a better search for highest weight elements in tensor products.
53ebc56	Created tensor product of crystals category.
81adbde	Implemented other tensor product categories.
aecbe54	Cleanup of commented out code.

[tscrim]

Commit: aecbe54

[tscrim]

Branch: public/crystals/highest_weight_tensor-16451

[tscrim]

Description changed:

--- 
+++ 
@@ -1 +1,5 @@
 Currently in a tensor product of crystals, we search through all elements and filter. However if given a highest weight element in a tensor product `a (x) b`, then `b` must be highest weight. This means we can use a backtracing algorithm to find all highest weight elements which this ticket implements.
+
+This first implements the categories of tensor products of crystals (currently all crystal subcategories are monoidal under tensor products).
+
+This also changes the output of the current `highest_weight_vectors()` to return a tuple since we don't want to allow modifications to the output of a `@cached_method`.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from aecbe54 to ed95e0e

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

6c33201	Merge branch 'public/crystals/highest_weight_tensor-16451' of trac.sagemath.org:sage into public/crystals/highest_weight_tensor-16451
ed95e0e	Made changes that Nicolas and I talked about.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

4f91c0f

Fix and test for Kashiwara convention of tensor products.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from ed95e0e to 4f91c0f

[tscrim]

comment:5

Hey Anne,

Nicolas looked at the category code and layout today and I made the necessary changes. The only thing needing review is the backtracing algorithm in highest_weight_crystals.py; would you be willing to finish the review.

Thanks,

Travis

[anneschilling]

Reviewer: Anne Schilling, Nicolas M. Thiery

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1 and set ticket back to needs_review. New commits:

c063d5b

Fixed typo

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 4f91c0f to c063d5b

[vbraun]

comment:9

sage -t --long src/doc/en/thematic_tutorials/lie/crystals.rst
**********************************************************************
File "src/doc/en/thematic_tutorials/lie/crystals.rst", line 701, in doc.en.thematic_tutorials.lie.crystals
Failed example:
    T.highest_weight_vectors()
Expected:
    [[1, 1, 1], [1, 2, 1], [2, 1, 1], [3, 2, 1]]
Got:
    ([1, 1, 1], [2, 1, 1], [1, 2, 1], [3, 2, 1])
**********************************************************************
1 item had failures:
   1 of 147 in doc.en.thematic_tutorials.lie.crystals
    [113 tests, 1 failure, 1.85 s]

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from c063d5b to c9df470

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

c9df470

Fixed trivial doctest failure due to new format.

[vbraun]

Changed branch from public/crystals/highest_weight_tensor-16451 to c9df470

[kcrisman]

Changed reviewer from Anne Schilling, Nicolas M. Thiery to Anne Schilling, Nicolas M. Thiéry

[kcrisman]

Changed commit from c9df470 to none