Ticket #14141: trac_14141-BK-fix-fs.patch

sage/combinat/knutson_tao_puzzles.py

@@ -906 +906 @@
 
 def BK_pieces(max_letter):
     """
-    Defines the puzzle pieces used in computing the Belkale-Kumar coefficients for any partial flag variety in type `A`.
+    The puzzle pieces used in computing the Belkale-Kumar coefficients for any
+    partial flag variety in type `A`.
+
+    There are two types of puzzle pieces:
+
+    - a triangle, with each edge labeled with the same letter;
+    - a rhombus, with edges labeled `i`, `j`, `i`, `j` in clockwise order with
+      `i > j`.
+
+    Each of these is rotated by 60 degrees, but not reflected.
+
+    We model the rhombus pieces as two triangles: a delta piece north-west
+    label `i`, north-east label `j` and south label `i(j)`; and a nabla piece
+    with south-east label `i`, south-west label `j` and north label `i(j)`.
 
     INPUT:
 
-    - ``max_letter`` -- positive integer specifying the number of steps in the partial flag variety
+    - ``max_letter`` -- positive integer specifying the number of steps in the
+      partial flag variety, equivalently, the number of elements in the
+      alphabet for the edge labels. The smallest label is `1`.
 
     REFERENCES:
 
-    .. [KnutsonPurbhoo10] A. Knutson, K. Purbhoo, Product and puzzle formulae for `GL_n` Belkale-Kumar coefficients,
-       {{{:arXiv:`1008.4979`}}}
+    .. [KnutsonPurbhoo10] A. Knutson, K. Purbhoo, Product and puzzle formulae
+       for `GL_n` Belkale-Kumar coefficients, {{{:arXiv:`1008.4979`}}}
 
     EXAMPLES::
 
         sage: from sage.combinat.knutson_tao_puzzles import BK_pieces
         sage: BK_pieces(3)
-        Nablas : [0\0/0, 1\1/1, 2\2/2, 3\3/3]
-        Deltas : [0/0\0, 1/1\1, 2/2\2, 3/3\3]
+        Nablas : [1\1/1, 1\2(1)/2, 1\3(1)/3, 2(1)\2/1, 2\1/2(1), 2\2/2, 2\3(2)/3, 3(1)\3/1, 3(2)\3/2, 3\1/3(1), 3\2/3(2), 3\3/3]
+        Deltas : [1/1\1, 1/2\2(1), 1/3\3(1), 2(1)/1\2, 2/2(1)\1, 2/2\2, 2/3\3(2), 3(1)/1\3, 3(2)/2\3, 3/3(1)\1, 3/3(2)\2, 3/3\3]
+
     """
-    max_letter += 1
-    forbidden_border_labels = ['%s%s' % (i,j) for j in range(max_letter-1) for i in range(j+1,max_letter)]
+    forbidden_border_labels = ['%s(%s)' % (i, j)
+                                for i in range(1, max_letter + 1)
+                                for j in range(1, i)]
     pieces = PuzzlePieces(forbidden_border_labels)
-    for j in range(0, max_letter):
-        jstr = '%s'%j
-        pieces.add_piece(DeltaPiece(jstr,jstr,jstr), rotations = 60)
-
-    for i in range(j+1, max_letter):
-        istr = '%s'%i
-        pieces.add_piece(DeltaPiece(istr+jstr, istr, jstr), rotations = 60)
-
+    for i in range(1, max_letter + 1):
+        piece = DeltaPiece('%s'%i, '%s'%i, '%s'%i)
+        pieces.add_piece(piece, rotations=60)
+        for j in range(1, i):
+            piece = DeltaPiece(north_west='%s'%i, north_east='%s'%j, south='%s(%s)'%(i,j))
+            pieces.add_piece(piece, rotations=60)
     return pieces
 
 class PuzzleFilling(object):
@@ -1478 +1493 @@
             {(1, 2): 0/\21  21\/0, (1, 3): 2/\2  21\/1, (3, 3): 0/0\0, (4, 5): 2/\1  1\/2, (4, 4): 1/1\1, (5, 5): 2/2\2, (1, 4): 1/\1  2\/21, (1, 1): 1/2\21, (1, 5): 2/\1  1\/2, (2, 3): 1/\1  10\/0, (2, 2): 0/1\10, (2, 5): 2/\2  2\/2, (3, 4): 1/\0  0\/1, (2, 4): 21/\2  1\/1, (3, 5): 2/\0  0\/2}]
 
 
-        Belkale-Kumar puzzles::
+        Belkale-Kumar puzzles (the following example is Figure 2 of [KnutsonPurbhoo10])::
 
-            sage: ps = KnutsonTaoPuzzleSolver("BK", 3)
-            sage: solns = ps('10212', '12012')
-            sage: sorted(solns, key=str)
+            sage: ps = KnutsonTaoPuzzleSolver('BK', 3)
+            sage: solns = ps('12132', '23112')
+            sage: len(solns)
+            1
+            sage: solns[0].south_labels()
+            ('3', '2', '1', '2', '1')
+            sage: solns
+            [{(1, 2): 2/\3(2)  3(1)\/1, (1, 3): 1/\3(1)  3(2)\/2, (3, 3): 1/1\1, (4, 5): 1/\1  2\/2(1), (4, 4): 2/2\2, (5, 5): 2(1)/1\2, (1, 4): 3/\3  3(1)\/1, (1, 1): 1/3\3(1), (1, 5): 2/\2  3\/3(2), (2, 3): 2/\2  2(1)\/1, (2, 2): 1/2\2(1), (2, 5): 3(2)/\3  2(1)\/1, (3, 4): 2/\1  1\/2, (2, 4): 1/\2(1)  2\/2, (3, 5): 1/\1  1\/1}]
 
         TESTS:
 
@@ -1576 +1596 @@
         """
         return "Knutson-Tao puzzle solver with pieces:\n%s" % self._puzzle_pieces
 
+    def puzzle_pieces(self):
+        r"""
+        The puzzle pieces used for filling in the puzzles.
+
+        EXAMPLES::
+
+            sage: from sage.combinat.knutson_tao_puzzles import KnutsonTaoPuzzleSolver
+            sage: ps = KnutsonTaoPuzzleSolver('H')
+            sage: ps.puzzle_pieces()
+            Nablas : [0\0/0, 0\10/1, 10\1/0, 1\0/10, 1\1/1]
+            Deltas : [0/0\0, 0/1\10, 1/10\0, 1/1\1, 10/0\1]
+        """
+        return self._puzzle_pieces
+
     def _fill_piece(self, nw_label, ne_label, pieces):
         """
         Fillings of a piece.