Английская Википедия: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 Шаблон:Harvtxt 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:
- P is graded and locally finite with a unique minimal element;
- for every two distinct elements x, y of P, the number of elements covering both x and y is the same as the number of elements covered by both x and y; and
- for every element x of P, the number of elements covering x is exactly r more than the number of elements covered by x.
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 = r I (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 Шаблон:Nowrap distributive lattice, while Шаблон:Harvtxt showed that these are the only Шаблон:Nowrap 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. Шаблон:Harvtxt 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 Шаблон:Harvtxt, where it is shown that there are uncountably many Шаблон:Nowrap. On the other hand, explicit examples of differential posets are rare; Шаблон:Harvtxt 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 Шаблон:Nowrap and Шаблон:Nowrap 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 Шаблон:Harvtxt that if P is a differential poset with Шаблон:Math vertices at rank n, then
- <math>p(n) \le r_n \le F_n,</math>
where p(n) is the number of integer partitions of n and Шаблон:Math is the nth 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 Шаблон:Harvtxt, while the lower bound remains open. Шаблон:Harvtxt proved an asymptotic version of the lower bound, showing that
- <math> r_n \gg n^a \exp(2\sqrt{n}) </math>
for every differential poset and some constant a. By comparison, the partition function has asymptotics
- <math> p(n) \sim \frac{1}{4n\sqrt{3}} \exp\left({\pi \sqrt {\frac{2n}{3}}}\right).</math>
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 Шаблон:Harvtxt showed that the rank sizes of an Шаблон:Nowrap 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:
- The number of paths of length 2n in the Hasse diagram of P beginning and ending at the minimal element is Шаблон:Math, where Шаблон:Math is the double factorial function. In an Шаблон:Nowrap poset, the number of such paths is Шаблон:Math.Шаблон:Sfn
- The number of paths of length 2n in the Hasse diagram 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 Шаблон:Math. In an Шаблон:Nowrap poset, the number is Шаблон:Math.Шаблон:Sfn
- The number of upward paths of length n in the Hasse diagram of P beginning with the minimal element is equal to the number of involutions in the symmetric group on n letters. In an Шаблон:Nowrap poset, the sequence of these numbers has exponential generating function Шаблон:Math.Шаблон:Sfn
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 Шаблон:Harvtxt. 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; Шаблон:Harvtxt 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.Шаблон:Citation needed In another direction, Шаблон:Harvtxt defined dual graded graphs corresponding to any Kac–Moody algebra.
Other variations are possible; Шаблон:Harvtxt defined versions in which the number r in the definition varies from rank to rank, while Шаблон:Harvtxt defined a signed analogue of differential posets in which cover relations may be assigned a "weight" of −1.
References
Sources
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation (Harvard College undergraduate thesis)
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
