Differential poset: Difference between revisions

Revision as of 21:39, 22 October 2013

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In [[mathematics]], a '''differential poset''' is a [[partially ordered set|poset]] (or ''poset'' for short) satisfying certain local properties. (The formal definition is given below.) TheseThis family of posets werewas introduced by {{harvtxt|Stanley|1988}} as a generalization of [[Young's lattice]] (the poset of [[integer partition]]s ordered by inclusion), many of whose [[combinatorics|combinatorial]] properties are shared by all differential posets. In addition to Young's lattice, the other most significant example of a differential poset is the [[Young-Fibonacci lattice]].

==Definitions==

A poset ''P'' is said to be a differential poset, and in particular to be ''r''-differential (where ''r'' is a positive integer), if it satisfies the following conditions:

* ''P'' is [[graded poset|graded]] and [[locally finite poset|locally finite]] with a unique minimal element;

* for every two distinct elements ''x'', ''y'' of ''P'', the number of elements [[covering relation|covering]] both ''x'' and ''y'' is the same as the number of elements covered by both ''x'' and ''y''; and

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The defining properties may also be restated in the following [[linear algebra]]ic setting: taking the elements of the poset ''P'' to be formal [[basis (linear algebra)|basis]] vectors of an (infinite dimensional) [[vector space]], let ''D'' and ''U'' be the [[linear operator|operators]] defined so that ''D'' ''x'' is equal to the sum of the elements covered by ''x'', and ''U'' ''x'' is equal to the sum of the elements covering ''x''. (The operators ''D'' and ''U'' are called the ''down'' and ''up operator'', for obvious reasons.) Then the second and third conditions may be replaced by the statement that ''DU'' – ''UD'' = ''rI'' (where ''I'' is the identity).

(This latter reformulation makes a differential poset into a combinatorial realization of a [[Weyl algebra]], and in particular explains the name ''diferential'': the operators "''d''/''dx''" and "multiplication by ''x''" on the vector space of polynomials obey the same commutation relation as ''U'' and ''D''/''r''.)

==Examples==

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==History, significance, and open questions==

==Properties==

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Every differential poset shares a large number of combinatorial properties. A few of these include:

==Properties==

{{expand section|date=October 2013}}

Every differential poset ''P'' shares a large number of combinatorial properties. A few of these include:

* The number of paths of 2''n'' + 1 elements of ''P'' beginning with the minimal element such that the first ''n'' steps are covering relations from a smaller to a larger element of ''P'' while the last ''n'' steps are covering relations from a larger to a smaller element of ''P'' is ''n''!.

==References==