Implementation of energy function for crystals

[anneschilling]

This patch implements the energy function and affine grading for affine crystals and adds an
entry in the corresponding thematic tutorial. It also adds functions to obtain the level and test perfectness
for Kirillov-Reshetikhin crystals.

In addition, it removes the condition that for the Lusztig involution the element is supposed
to live in a crystal with a single highest weight element, and modifies the methods
to_highest_weight and to_lowest_weight in crystal.py to make them more robust.

The update to the thematic tutorial can be built using

sage -docbuild thematic_tutorials html

is available here:

http://www.math.ucdavis.edu/~anne/sage/lie/affine_crystals.html

Apply: attachment: trac_11546-crystals_energy-as.patch

Depends on #11183

CC: @sagetrac-sage-combinat

Component: combinatorics

Keywords: energy, crystals

Author: Anne Schilling

Reviewer: Tom Denton

Merged: sage-4.7.2.alpha1

Issue created by migration from https://trac.sagemath.org/ticket/11546

[anneschilling]

Description changed:

--- 
+++ 
@@ -1 +1,11 @@
 This patch implements the energy function for affine crystals and adds an entry in the corresponding thematic tutorial.
+
+The update to the thematic tutorial can be built using
+
+```
+sage -docbuild thematic_tutorials html
+```
+
+is available here:
+
+ http://www.math.ucdavis.edu/~anne/sage/lie/affine_crystals.html 

[anneschilling]

comment:2

Ready for review!

[anneschilling]

Description changed:

--- 
+++ 
@@ -1,4 +1,7 @@
-This patch implements the energy function for affine crystals and adds an entry in the corresponding thematic tutorial.
+This patch implements the energy function and affine grading for affine crystals and adds an entry in the corresponding thematic tutorial.
+
+In addition, it removes the condition that for the Lusztig involution the element is supposed to live in a crystal with a single highest weight element, and modifies the methods
+to_highest_weight and to_lowest_weight in crystal.py to make them more robust.
 
 The update to the thematic tutorial can be built using
 

[sdenton4]

comment:4

1. Funny problem with the classical crystal command, unrelated to patch?
   K.classical_decomposition seems to work fine, though.

    sage: K = KirillovReshetikhinCrystal(['A',2,1],2,1) 
    sage: Kc=K.classical_crystal()
    sage: Kc
    ERROR: An unexpected error occurred while tokenizing input
    The following traceback may be corrupted or invalid
    The error message is: ('EOF in multi-line statement', (396, 0))

    ---------------------------------------------------------------------------
    AttributeError                            Traceback (most recent call last)
    ....
    AttributeError: 'int' object has no attribute 'value'

2. small request... Could there be a 'level' function on KR crystals? It looks like this would be as easy as wrapping the assertion in the energy function in a separate function, and would give a natural response to the user when the 'All crystals in the tensor product need to be perfect of the same level' assertion error comes up.

3. The 'list' option removed from the 'to_highest_weight' functions looks like it was a bit of extraneous code, so this is good!
   I'm a little concerned though, because the old way returned:

        return self.to_highest_weight(list = list + [i], index_set = index_set)

While the new returns:

        hw = self.to_highest_weight(index_set = index_set) 
        return [hw[0], [i] + hw[1]]

It looks like the new 'i' is being added to opposite ends of the list in these two cases; is this consistent one way or the other across the code-base?

This reminds me of a functionality extension I had thought would be nice at one point, namely being able to feed a list to the 'e' and 'f' functions instead of just a single letter. It would both extend the functionality a bit and establish a convention for which way this 'list' should be applied!

[anneschilling]

comment:5

Hi Tom,

Thank you for looking at the patch!

Replying to @sdenton4:

1. Funny problem with the classical crystal command, unrelated to patch?
   K.classical_decomposition seems to work fine, though.

    sage: K = KirillovReshetikhinCrystal(['A',2,1],2,1) 
    sage: Kc=K.classical_crystal()
    sage: Kc
    ERROR: An unexpected error occurred while tokenizing input
    The following traceback may be corrupted or invalid
    The error message is: ('EOF in multi-line statement', (396, 0))

    ---------------------------------------------------------------------------
    AttributeError                            Traceback (most recent call last)
    ....
    AttributeError: 'int' object has no attribute 'value'

classical_crystal is an attribute, not a method, so the correct way of using it is

    sage: K = KirillovReshetikhinCrystal(['A',2,1],2,1)
    sage: Kc = K.classical_crystal
    sage: Kc
    The crystal of tableaux of type ['A', 2] and shape(s) [[1, 1]]

2. small request... Could there be a 'level' function on KR crystals? It looks like this would be as easy as wrapping the assertion in the energy function in a separate function, and would give a natural response to the user when the 'All crystals in the tensor product need to be perfect of the same level' assertion error comes up.

The notion of level really only exists for perfect crystals. However, the method affine_grading also works in the non-perfect setting. Note that in the assert there is no ceiling for the quotient!

3. The 'list' option removed from the 'to_highest_weight' functions looks like it was a bit of extraneous code, so this is good!
   I'm a little concerned though, because the old way returned:

        return self.to_highest_weight(list = list + [i], index_set = index_set)

While the new returns:

        hw = self.to_highest_weight(index_set = index_set) 
        return [hw[0], [i] + hw[1]]

Doing it this way gives the same output as before! I did not change any of the tests and they still pass.

It looks like the new 'i' is being added to opposite ends of the list in these two cases; is this consistent one way or the other across the code-base?

This reminds me of a functionality extension I had thought would be nice at one point, namely being able to feed a list to the 'e' and 'f' functions instead of just a single letter. It would both extend the functionality a bit and establish a convention for which way this 'list' should be applied!

This already exists and is used as follows:

    sage: K = KirillovReshetikhinCrystal(['A',2,1],1,1)
    sage: t = K(rows=[[1]])
    sage: t.f_string([1,2])
    [[3]]