#14641: Does the "promotion" method for tableaux really compute Schuetzenberger
promotion?
-----------------------------+----------------------------------------------
Reporter: darij | Owner: tbd
Type: defect | Status: new
Priority: major | Milestone: sage-5.10
Component: PLEASE CHANGE | Keywords: tableaux, partitions,
combinat, jeu de taquin, promotion
Work issues: | Report Upstream: N/A
Reviewers: | Authors:
Merged in: | Dependencies:
Stopgaps: |
-----------------------------+----------------------------------------------
Both {{{promotion}}} and {{{promotion_inverse}}} methods in
{{{combinat/tableau.py}}} take two inputs: the tableau {{{self}}} and an
integer {{{n}}}.
What exactly does the {{{n}}} do? While {{{promotion}}} has some kind of
docstring, I fail to understand it. The notion of promotion has an
unfortunate wealth of different meanings, but I can't recognize any of
them in what Sage computes. I expected the {{{n}}} to be the {{{i}}} in
the {{{\delta_i}}} of Ayyer-Klee-Schilling
http://arxiv.org/pdf/1205.7074v2.pdf (identifying standard Young tableaux
with saturated chains on the partition lattice), but then I would expect
that for {{{X}}} being a standard tableau, {{{X.promotion(n)}}} would
still be a standard tableau. This is not the case:
{{{
sage: X = StandardTableau([[1,2],[3,4],[5]])
sage: X.promotion(1)
[[1, 2], [4, 5], [6]]
}}}
It seems to me that {{{X.promotion(n-1)}}}, where {{{n}}} is the size of
the standard tableau {{{X}}}, computes the good old Schuetzenberger
promotion of {{{X}}}; but I am not quite sure and this seems to contradict
the docstring.
When {{{X}}} is rectangular, the code works very differently and some
completely strange things happen:
{{{
sage: S = StandardTableau([[1,3,5,6],[2,4,7,8]])
sage: S.promotion(0) # This should make perfect sense going by the
docstring...
---------------------------------------------------------------------------
IndexError Traceback (most recent call
last)
<ipython-input-95-23c3eab7210b> in <module>()
----> 1 S.promotion(Integer(0)) # This should make perfect sense going
by the docstring...
/home/darij/sage-5.10.beta2/local/lib/python2.7/site-
packages/sage/combinat/tableau.pyc in promotion(self, n)
1779 t = self.rotate_180()
1780 t = [[n+2-i for i in row] for row in t.to_list()]
-> 1781 t = Tableau(t).promotion_inverse(n)
1782 t = [[n+2-i for i in row] for row in t.to_list()]
1783 return Tableau(t).rotate_180()
/home/darij/sage-5.10.beta2/local/lib/python2.7/site-
packages/sage/combinat/tableau.pyc in promotion_inverse(self, n)
1742 if l<s:
1743 for i in range(l):
-> 1744 t[len(t)-1].append(n+1)
1745 else:
1746 t.append([n+1 for i in range(s)])
IndexError: list index out of range
sage: S.promotion(1)
[[1, 2]]
sage: S.promotion(2)
---------------------------------------------------------------------------
ValueError Traceback (most recent call
last)
<ipython-input-97-6a4133c5e025> in <module>()
----> 1 S.promotion(Integer(2))
/home/darij/sage-5.10.beta2/local/lib/python2.7/site-
packages/sage/combinat/tableau.pyc in promotion(self, n)
1779 t = self.rotate_180()
1780 t = [[n+2-i for i in row] for row in t.to_list()]
-> 1781 t = Tableau(t).promotion_inverse(n)
1782 t = [[n+2-i for i in row] for row in t.to_list()]
1783 return Tableau(t).rotate_180()
/home/darij/sage-5.10.beta2/local/lib/python2.7/site-
packages/sage/combinat/tableau.pyc in promotion_inverse(self, n)
1745 else:
1746 t.append([n+1 for i in range(s)])
-> 1747 return Tableau(t)
1748
1749 def promotion(self, n):
/home/darij/sage-5.10.beta2/local/lib/python2.7/site-
packages/sage/misc/classcall_metaclass.so in
sage.misc.classcall_metaclass.ClasscallMetaclass.__call__
(sage/misc/classcall_metaclass.c:1208)()
/home/darij/sage-5.10.beta2/local/lib/python2.7/site-
packages/sage/combinat/tableau.pyc in __classcall_private__(self, t)
258
259 if not map(len,t) in sage.combinat.partition._Partitions:
--> 260 raise ValueError("A tableau must be a list of lists of
weakly decreasing length.")
261
262 return Tableaux_all().element_class(Tableaux_all(), t)
ValueError: A tableau must be a list of lists of weakly decreasing length.
sage: S.promotion(3)
[[1, 2], [3, 4]]
sage: S.promotion(4)
---------------------------------------------------------------------------
ValueError Traceback (most recent call
last)
<ipython-input-99-9acbfaebd18c> in <module>()
----> 1 S.promotion(Integer(4))
/home/darij/sage-5.10.beta2/local/lib/python2.7/site-
packages/sage/combinat/tableau.pyc in promotion(self, n)
1779 t = self.rotate_180()
1780 t = [[n+2-i for i in row] for row in t.to_list()]
-> 1781 t = Tableau(t).promotion_inverse(n)
1782 t = [[n+2-i for i in row] for row in t.to_list()]
1783 return Tableau(t).rotate_180()
/home/darij/sage-5.10.beta2/local/lib/python2.7/site-
packages/sage/combinat/tableau.pyc in promotion_inverse(self, n)
1745 else:
1746 t.append([n+1 for i in range(s)])
-> 1747 return Tableau(t)
1748
1749 def promotion(self, n):
/home/darij/sage-5.10.beta2/local/lib/python2.7/site-
packages/sage/misc/classcall_metaclass.so in
sage.misc.classcall_metaclass.ClasscallMetaclass.__call__
(sage/misc/classcall_metaclass.c:1208)()
/home/darij/sage-5.10.beta2/local/lib/python2.7/site-
packages/sage/combinat/tableau.pyc in __classcall_private__(self, t)
258
259 if not map(len,t) in sage.combinat.partition._Partitions:
--> 260 raise ValueError("A tableau must be a list of lists of
weakly decreasing length.")
261
262 return Tableaux_all().element_class(Tableaux_all(), t)
ValueError: A tableau must be a list of lists of weakly decreasing length.
sage: S.promotion(5)
[[1, 2, 4], [3, 5, 6]]
sage: S.promotion(6)
---------------------------------------------------------------------------
TypeError Traceback (most recent call
last)
<ipython-input-101-ea0436786e82> in <module>()
----> 1 S.promotion(Integer(6))
/home/darij/sage-5.10.beta2/local/lib/python2.7/site-
packages/sage/combinat/tableau.pyc in promotion(self, n)
1781 t = Tableau(t).promotion_inverse(n)
1782 t = [[n+2-i for i in row] for row in t.to_list()]
-> 1783 return Tableau(t).rotate_180()
1784 p = self
1785 for c in self.cells_containing(n+1):
/home/darij/sage-5.10.beta2/local/lib/python2.7/site-
packages/sage/combinat/tableau.pyc in rotate_180(self)
1154 """
1155 if not self.is_rectangular():
-> 1156 raise TypeError, "the tableau must be rectangular to
use verticl_flip()"
1157
1158 return Tableau([ [l for l in reversed(row)] for row in
reversed(self) ])
TypeError: the tableau must be rectangular to use verticl_flip()
sage: S.promotion(7)
[[1, 2, 4, 7], [3, 5, 6, 8]]
sage: S.promotion(8) # even fails, odd works?
[[2, 4, 6, 7], [3, 5, 8, 9]]
}}}
The notion of promotion suffers from a wealth of different meanings, but
what I am seeing here doesn't match any I know...
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/14641>
Sage <http://www.sagemath.org>
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