Differential poset

In mathematics, a differential poset is a partially ordered set (or "poset" for short) satisfying certain local properties. (The formal definition is given below.) This family of posets was introduced by as a generalization of Young's lattice (the poset of integer partitions ordered by inclusion), many of whose 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:
These basic properties may be restated in various ways. For example, Stanley shows that the number of elements covering two distinct elements "x" and "y" of a differential poset is always either 0 or 1, so the second defining property could be altered accordingly.
The defining properties may also be restated in the following linear algebraic setting: taking the elements of the poset "P" to be formal basis vectors of an (infinite dimensional) vector space, let "D" and "U" be the 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 "differential": 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.
The canonical examples of differential posets are Young's lattice, the poset of integer partitions ordered by inclusion, and the Young–Fibonacci lattice. Stanley's initial paper established that Young's lattice is the only 1-differential distributive lattice, while showed that these are the only 1-differential lattices.
There is a canonical construction (called "reflection") of a differential poset given a finite poset that obeys all of the defining axioms below its top rank. (The Young–Fibonacci lattice is the poset that arises by applying this construction beginning with a single point.) This can be used to show that there are infinitely many differential posets. includes a remark that "[David] Wagner described a very general method for constructing differential posets which make it unlikely that [they can be classified]." This is made precise in , where it is shown that there are uncountably many 1-differential posets. On the other hand, explicit examples of differential posets are rare; gives a convoluted description of a differential poset other than the Young and Young–Fibonacci lattices.
The Young-Fibonacci lattice has a natural "r"-differential analogue for every positive integer "r". These posets are lattices, and can be constructed by a variation of the reflection construction. In addition, the product of an "r"-differential and "s"-differential poset is always an ("r" + "s")-differential poset. This construction also preserves the lattice property. It is not known for any "r" > 1 whether there are any "r"-differential lattices other than those that arise by taking products of the Young–Fibonacci lattices and Young's lattice.
Rank growth.
In addition to the question of whether there are other differential lattices, there are several long-standing open problems relating to the rank growth of differential posets. It was conjectured in that if "P" is a differential poset with vertices at rank "n", then
where "p"("n") is the number of integer partitions of "n" and is the "n"th Fibonacci number. In other words, the conjecture states that at every rank, every differential poset has a number of vertices lying between the numbers for Young's lattice and the Young-Fibonacci lattice. The upper bound was proved in . The lower bound remains open. proved an asymptotic version of the lower bound, showing that
for every differential poset and some constant "a". By comparison, the partition function has asymptotics
All known bounds on rank sizes of differential posets are quickly growing functions. In the original paper of Stanley, it was shown (using eigenvalues of the operator "DU") that the rank sizes are weakly increasing. However, it took 25 years before showed that the rank sizes of an "r"-differential poset strictly increase (except trivially between ranks 0 and 1 when "r" = 1).
Properties.
Every differential poset "P" shares a large number of combinatorial properties. A few of these include:
Generalizations.
In a differential poset, the same set of edges is used to compute the up and down operators "U" and "D". If one permits different sets of up edges and down edges (sharing the same vertex sets, and satisfying the same relation), the resulting concept is the "dual graded graph", initially defined by . One recovers differential posets as the case that the two sets of edges coincide.
Much of the interest in differential posets is inspired by their connections to representation theory. The elements of Young's lattice are integer partitions, which encode the representations of the symmetric groups, and are connected to the ring of symmetric functions; defined algebras whose representation is encoded instead by the Young–Fibonacci lattice, and allow for analogous constructions such as a Fibonacci version of symmetric functions. It is not known whether similar algebras exist for every differential poset. In another direction, defined dual graded graphs corresponding to any Kac–Moody algebra.
Other variations are possible; defined versions in which the number "r" in the definition varies from rank to rank, while defined a signed analogue of differential posets in which cover relations may be assigned a "weight" of −1.