Implementation of Knutson-Tao puzzles

[anneschilling]

This patch implements Knutson-Tao puzzles in various settings (Grassmannian, equivariant Grassmannian, K theoretic, Belkale-Kumar puzzles).

This code was written during Sage Days 45 at ICERM with Franco Saliola, Anne Schilling, and Avinash Dalal in discussions with Allen Knutson.
The code was tested afterwards by Liz Beazley and Ed Richmond.

Apply:

• attachment: trac_14141-knutson_tao_puzzles-as.patch

Depends on #14299

CC: @sagetrac-sage-combinat @saliola

Component: combinatorics

Keywords: puzzles, Littlewood-Richardson coefficiens, days45

Author: Franco Saliola, Avinash Dalal, Anne Schilling

Reviewer: Allen Knutson, Liz Beazley, Ed Richmond

Merged: sage-5.10.beta4

Issue created by migration from https://trac.sagemath.org/ticket/14141

[anneschilling]

Changed author from Franco Saliola, Allen Knutson, Avinash Dalal, Anne Schilling to Franco Saliola, Avinash Dalal, Anne Schilling

[anneschilling]

Reviewer: Allen Knutson, Liz Beazley, Ed Richmond

[anneschilling]

Description changed:

--- 
+++ 
@@ -1 +1,4 @@
+This patch implements Knutson-Tao puzzles in various settings (Grassmannian, equivariant Grassmannian, K theoretic, Belkale-Kumar puzzles).
 
+This code was written during Sage Days 45 at ICERM with Franco Saliola, Anne Schilling, and Avinash Dalal in discussions with Allen Knutson.
+The code was tested afterwards by Liz Beazley and Ed Richmond (who also contributed to the bijection between puzzles and tableaux).

[anneschilling]

comment:2

Hi Franco,

The patch is basically ready to go now. Could you please look over my changes? Also, there is one method which is currently broken (namely _plot_color_edges), which would still be good to have around.

Thanks!

Anne

[anneschilling]

comment:3

Added documentation to rst file.

Anne

[saliola]

comment:4

What version is this patch based on ? (It doesn't apply to sage-5.8.)

[anneschilling]

comment:5

Replying to @saliola:

What version is this patch based on ? (It doesn't apply to sage-5.8.)

For me it applies to sage-5.9.beta2.

[fchapoton]

comment:6

some stupid suggestions:

• use .. TODO instead of just todo

• use the role :arxiv:`xxxx.xxxx` when citing a paper on arxiv

[anneschilling]

comment:7

Frédéric, thank you for the comments. They should be fixed now. Currently, a more serious issue is the following that Liz observed:

sage: Hps = PuzzleSolver("H")
sage: Hsolns = Hps('001001', '001010')
sage: Puzz = Hsolns[0]
sage: Puzz.LR_tab_from_puzzle()
---------------------------------------------------------------------------
AssertionError                            Traceback (most recent call last)
<ipython-input-39-2284d874276b> in <module>()
----> 1 Puzz.LR_tab_from_puzzle()

/Applications/sage-5.9.beta2/local/lib/python2.7/site-packages/sage/combinat/knutson_tao_puzzles.pyc in LR_tab_from_puzzle(self)
   1629                 for j in range(0,lam[k-1]):
   1630                     LRrow.append(None)
-> 1631                 LRrow.extend(self.path_thru_puzzle(i+1))
   1632                 LRtableaux.append(LRrow)
   1633         return SkewTableau(LRtableaux)

/Applications/sage-5.9.beta2/local/lib/python2.7/site-packages/sage/combinat/knutson_tao_puzzles.pyc in path_thru_puzzle(self, coord)
   1565         """
   1566         i = coord
-> 1567         assert self._ne_labels[coord-1] == '1', "The coordinate needs to be a coordinate of a 1 on the north east boundary"
   1568         j = self._n
   1569         k = 0

AssertionError: The coordinate needs to be a coordinate of a 1 on the north east boundary