Differential poset

In mathematics, a differential poset is a poset satisfying certain local properties. (The formal definition is given below.) These posets were introduced by Stanley (1988) as a generalization of Young's lattice (the poset of integer partitions ordered by inclusion). 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 = 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.)

Properties

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

References

• Stanley, Richard P. (1988), "Differential posets", Journal of the American Mathematical Society, 1 (4), American Mathematical Society: 919–961, doi:10.2307/1990995, JSTOR 1990995