Implementation of the Schuetzenberger involution, the Lusztig involution and the promotion operator

[sagetrac-ebeyerstedt]

This patch implements the Schuetzenberger involution on both words and tableaux. In addition it generalizes the promotion operator on tableaux to act not only on rectangular tableaux, but tableaux of any shape.

For crystals in the category ClassicalCrystals, the Lusztig involution is implemented and tested in type A to give the same result as the Schuetzenberger involution.

A section about the Lusztig involution is added to the thematic tutorial on crystals. A copy of the html documentation built using

sage -docbuild thematic_tutorials html

is available here:

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

(which might not always be completely up to date).

Apply trac_10446_schuetzenberger_involution-as.patch

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

Component: combinatorics

Keywords: schuetzenberger, words, crystals

Author: Erin Beyerstedt, Ilke Canakci, Anne Schilling

Reviewer: Jason Bandlow, Nicolas M. Thiéry

Merged: sage-4.7.1.alpha0

Issue created by migration from https://trac.sagemath.org/ticket/10446

[anneschilling]

comment:2

This already looks very good!

One suggestion I have is to check whether there is an alphabet associated to a given word 'self'. In this case it would be better to use the maximum of the alphabet instead of the maximum of the appearing letters. For example, if you inputted

sage: w=Word([1,2,3],alphabet=[1,2,3,4,5])
sage: w.alphabet()
Ordered Alphabet [1, 2, 3, 4, 5]

then you would like the Schuetzenberger involution to return the word
345 instead of 123.

Anne

[sagetrac-ebeyerstedt]

Implementation of Schuetzenberger Involution on Words and Tableau

[sagetrac-ebeyerstedt]

comment:6

Attachment: trac_10446_schuetzenberger_involution.patch.gz

This patch now contains two functions, both of which define the schuetzenberger involution - one that works on the class of words and the other on tableau.

[sagetrac-ebeyerstedt]

Description changed:

--- 
+++ 
@@ -1 +1 @@
-Implementation of the schuetzenberger involution on words.
+Implementation of the schuetzenberger involution on both words and tableau.

[anneschilling]

Reviewer: Anne Schilling

[anneschilling]

Description changed:

--- 
+++ 
@@ -1 +1 @@
-Implementation of the schuetzenberger involution on both words and tableau.
+Implementation of the schuetzenberger involution on both words and tableaux.

[anneschilling]

comment:8

Replying to @anneschilling:

This looks good and all tests pass! One suggestion I have is that it should be possible to pass the alphabet underlying a given tableau for the method schuetzenberger_involution for tableaux. Currently

sage: t = Tableau([[1,1,4],[2]])
sage: t.schuetzenberger_involution()
[[1, 3, 4], [4]]

but if the underlying alphabet is different, the answer would be different.
This would require to change the method to_word in tableau.py.

Also, for words it would be good to add a test that shows that one can have repeated letters:

sage: w = Word([2,1,1,4], alphabet=[1,2,3,4,5,6])
sage: w.schuetzenberger_involution()
word: 3665

Thanks so much for your work on this!

Anne