Let P be a fixed poset. A split (I,J) of P is a decomposition
of P into an ideal I and a filter J which do not intersect. A
labelled split (I,J) also consists of:
A red numbering of an ideal I of P, which is an assignment of the red
numbers 1, 2, ..., |I| to the elements of I which respects the partial
order of P, and
A green numbering of a filter J of P, which is an assignment of the
green numbers 1, 2, ..., |J| to the elements of J which respects the
partial order of P.
Suppose that red and green numbers have been assigned to all of the elements of P. Then Move(k) is the interchange of this green number k with the largest of the red numbers which are assigned to elements which are covered by y.
Suppose that we have a labelled split (I,J) of P. Given a minimal element x of J, then Slide(k) is the process consisting of performing Move(k) as many times as possible: the green number k is repeatedly swapped with the largest of the red numbers that it covers until it covers no red numbers.
Suppose that we have a labelled split (I,J) of P. Then JDT[I,J] consists of a filter of P together with a red numbering which is produced as follows: Perform Slide(1), Slide(2), Slide(3), ... , until all of the green numbers have been slid out as far as possible. Then throw away all elements which are labelled with green numbers, leaving a filter of P whose elements are labelled with red numbers.
Definition: A poset P has the jeu de taquin property if:
For all splits (I,J) of P and
for all red numberings of I,
the result JDT[I,J] is independent of the green numbering chosen for
J.
Theorem 1 on our home page states that every d-complete poset has the jeu de taquin property.
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