104

数理解析研究所講究録
第 765 巻 1991 年 104-108

DIFFERENTIAL POSETS
Richard P. Stanley
Department of Mathematics
Massachusetts Institute of Technology
Cambrid ge, MA 02139 USA
e-mail: rstan@math.mit.edu
EXTENDED ABSTRACT
1. Definitions. Let be a positive integer. An r-differential poset is a partially ordered
set $P$ satisfying the following three axioms:$r$(D1) $P$ is locally finite with unique minimal element , and is graded (i.e., for any $x\in P$ ,all saturated chains between and : have the same length).
$\hat{0}$$\hat{0}$$’\iota$(D2) For any $x,$ $y\in P$ , if exactly
elements of $P$ cover both$k$(D3) If $x\in P$ covers

$k$

$k$

$x$

elements 01
and .

$l^{2}$are covered by both$x$and , then exactly
$y$

$y$

elements of $P,$

$l1_{1\langle}>J1^{\cdot}$is covered by $k+r$ elements of $P$ .

po.set. Let us note two simple
A poset which is r-differential for some is ci, $IIc(1$ aimplies that the integer of (D2) is or 1,properties of differential posets: (a) Axiomand (b) if $P$ is a lattice satisfying (D1) and (D3), ( ]}( (D2) is equivalent to modularity.
,/$rl^{J}i/\gamma C^{J}te\cdot ntial$$\cdot$$k$$(]^{-})1)$$0$$-\backslash n$$.$

2. Examples of differential posets. There are tw principal examples of l-differential
totlte set of all sequences
posets. The first is Young lattice}’’, (only finitely manyof nonnegative integers.
, ordered com]”of the integer
ponentwise. Thus the element of is ust, il(denoted]’$N\cross
N$. Equivalently, is isomorphic to befinile order ideals of, ordered byinclusion (where $N$ denotes the chain $0<1<\cdots$ ). ]‘ is the unique l-differential distributiveof all partitions of ), then the subposet
lattice. If denotes the ith level of ]“ ( : tlie
is the Bratteli diagram of the pai of algebras $(CS_{n}, CS_{7\iota+1})$ , where $CS_{m}$ denotes
the group algebra (over the complex nu mbers C) of the symmetric group. For this reason
related to the representation theory
many combinatorial and algcbraic properties of ]’ofdenotcs tlie number of saturated chains between and , then. For instance, if,,whereare just thethe(complex) representations of. Hence by well known results in represenlat ion theory,
$0$

$1$

$s$

($>.fi_{11}c^{s}c1$$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq 0$$\lambda$$|$$\lambda=(\lambda_{1}, \lambda_{2}, \ldots)$$\backslash vi1_{:}11$$j$$\int$$\lambda\vdash n)$) $e$$\lambda_{i}\neq 0$) $m\cdot/ltion$$n= \sum\lambda_{i}$$\backslash _{-};(11oI^{\cdot}$$t$$\cdot$$i.$ ( $!.$

$Y_{i}$$i$$\backslash s\cdot c^{s}\downarrow,$’$Y_{i}\cup Y_{i+1}$$1$$S_{m}$$\dot{\subset}\backslash \iota\cdot e$$S_{m}$$e(\lambda)s$$\hat{0}$$e(\lambda)$$\lambda\vdash\uparrow\tau$$dc^{J}g_{1\}}(()s’01^{\cdot}\{_{J}|_{1}\epsilon^{Y}i_{1}\cdot\iota\cdot ec1ncil-\supset le$$S_{n}$$\sum_{\lambda\vdash n}e(\lambda)=\#\{t\{\in S,-, |_{1l}^{\underline{\prime}}=1\}$/$\lambda$

105$\sum_{\lambda\vdash n}e(\lambda)^{2}=n!$The theory of differential posets shows that these formulae are consequences only of properties $(D1)-(D3)$ of Young’s lattice Y.
The second principal example of a l-differential poset is denoted or $Z(1)$ and is called
the Fibonacci l-differential poset. For the precise definition see [1], and for further combinatorial properties see [3]. is the unique l-differential lattice for which every complemented
interval has length at most two. The number of elements of of rank is the ith Fibonacciby the property that $Z_{n}\cup Z_{n+1}$ is thenumber . Define complex semisimple algebras, and it would be interesting to. ThenBratteli diagram of the pairfind a “nice” combinatorial definition of .
$Z$

$Z$

$Z$

$p_{i}$$F_{i}$$i$$\mathcal{F}_{n}$$\dim \mathcal{F}_{n}=n!$$(\mathcal{F}_{n}, \mathcal{F}_{n+1})$$\mathcal{F}_{n}$Conjecture. The only l-differential lattices are$Y$and$Z$.

3. The operators
and $D$ , and enumerative properties of differential posets.
andThe basic tools for investigating differential posets are two linear operators denoted$P$$D$ . Let $K$
be a field of characteristic . For any locally finite poset with such that every
element is covered by finitely many elements, let
be the vector space of all (infinite)$P$linear combinations of elements of . Define linear transformations $U,$ $D$ : $K^{P}arrow K^{P}$ by$U$$U$$\hat{0}$$0$$K^{P}$$U(x)= \sum_{\langle y\in C+x)}y$$D(x)= \sum_{y\in C^{-}\langle x)}y$,where $x\in P$ , and where $C^{+}(x)$ (respectively, $C^{-}(x)$ ) is the set of elements which coverby requiring(respectively, which covers). Moreover,and $D$ are extended to all ofthem to preserve infinite linear combinations.
$x$$K^{P}$$U$$x$

Theorem. The following two conditions are equivalent:$(a)$ DU–UD $=rI$ (where I denotes the identity$(b)P$ is r-differential.

)$ope\uparrow\cdot ator$

Proposition. Let$P$

be r-differential. Let $P=\Sigma_{x\in P}x$ . Then $UP=(D+r)P$ .

Thus a differential poset affords a representation of the Weyl algebra $C[x, d/dx]$ , whererepresents and $D/r$ represents $d/dx$ . This explains the terminology “differential poset.”
The commutation rule DU–UD $=rI$ leads to many formulae involving the counting ofcertain paths in the Hasse diagram of . We will just state a sample of these results here.
See Section 3 of [1] for further details.
$U$

$x$

$P$

2

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Theorem. Let $P$ be an r-differential poset.
$(a)$ Let $\alpha(0arrow n)$ denote the numberof saturated chains $\hat{0}=x_{0}<x_{1}<\cdots<x_{n}$ in, the set of elements of of rank ). Then(so$P$$x_{i}\in P_{i}$

$P$

$i$

$\sum_{n\geq 0}\alpha(0arrow n)\frac{t^{n}}{n!}=\exp(rt+\frac{1}{2}rt^{2})$.

Equivalently,$\alpha(0arrow n)=\sum_{w^{2}=1}r^{c(w)}$,summed over all involutions in , where $c(w)$ denotes the number of cycles of .
$(b)$ Let $\alpha(0arrow narrow 0)$ denote the numberof “Hasse walks “ $\hat{0}=x_{0}<x_{1}<\cdots<x_{n}>$$y_{n-1}>\cdots>y_{0}=\hat{0}$ (soand have rank ). Then
$S_{n}$$w$$x_{i}$$w$$i$$y_{i}$$\alpha(0arrow narrow 0)=r^{n}n!$Equivalently,$\sum_{x\in P_{n}}e(x)^{2}=r^{n}??!$,

to .
where $e(x)$ is the number of saturated chains in$(c)$ Letdenote the number of Hasse walks inof length
such that for all either
number of sequences. Then$Pfi^{\sim}om\hat{0}$

$x$

$P$

$\delta_{n}$$\hat{0}=x_{0},$$x_{1},$$\ldots,$$i$$x_{n}$$n$$x_{t}$$i.e$ , thebeginning atcovers or is covered by$\hat{0},$$x_{i-1}$$\sum_{n\geq 0}\delta_{n}\frac{t^{n}}{n!}=\exp(\uparrow\cdot t+rt^{2})$Let. Then$(d)$$\hat{0}$$\kappa_{2n}$denote the number of Hasse walks in$P$.

of length$\kappa_{2n}=1\cdot 3\cdot 5\cdots(2n-1)r^{n}$$2n$beginning and ending at.

4. Eigenvalues and eigenvectors. For certain linear transformations connected withandon a differential poset, we can explicitly compute their eigenvalues
the operatorsand eigenvectors. We state here the simplest results in this direction; see Section 4 of [1] forfurther results.
$D$$U$Theorem. Let
be an r-differential poset. Let
denote the linear transformation$UD$ restricted to the subspaceThenpolynomial (normalizedthecharacteristic.
ofto be monic) ofis given by$P$$UD_{j}$$K^{P_{j}}$$K^{P}$$UD_{j}$$\prod_{i=0}^{j}(\lambda-ri)^{p_{g-};-p_{y-\cdot-1}}$

3,

10?

wheregiven by$p_{i}=\neq P_{i}$. Moreover, the eigenvector$E_{j}$

corresponding to the largest eigenvalue $rj$ is$E_{j}= \sum_{x\in P_{\dot{J}}}e(x)x$

where $e(x)$ is the number of saturated chains from$\hat{0}$,to$x$.

There is also a recursive formula for the other eigenvectors oflattice $Y$ we can be more explicit about these other eigenvectors.

$UD_{j}$denote the irreducible character ofTheorem. Let. Then for any partition $\mu ofj$ the vector$\chi^{\lambda}$$S_{j}$. In the case of Young’s

corresponding to the partition$\lambda ofj$

$X_{\mu}= \sum_{\lambda\vdash j}\chi’\backslash (\mu)\lambda$

is an eigenvector for(the numbercorresponding to the eigenvalue:
Moreover,partsequal
rthogonal
tothe1).
eigenvectors
completegive’sasetofofof
an orthonormal basis).
(with respect to the scalar product which makesfor$K^{Y_{J}}arrow K^{Y_{J}}$$UD_{j}$$m_{1}(\mu)$$0$$X_{\mu}$$\mu$$UD_{j}$$Y_{j}$

5. Variations on differential posets. There are several ways to extend the notionof a differential poset and still retain some of the basic theory. Two of the most interesting

variations are the following.
be a sequence of integers. An r-differential poset is aVariation 1. Let$P$
poset satisfying axioms (D1) and (D2) above, together with
$r=(\uparrow 0, r_{1}, \ldots)$(D3’) If $x\in P_{j}$ covers$k$elements of$P$, then$x$is covered by $k+r_{j}$ elements of$P$.

A poset which is r-differential for some is called sequentially differential. There are many
more interesting examples of sequentially differential posets than of just differential posets.
For instance, the boolean algebra $B_{n}=2^{n}$ , as well as a productof three-element chains,is sequentially differential. All the properties of differential posets discussed above carryover to the sequential case, though the statements of the results are often more complicated(since they involve infinitely many variables. rather than just the single variable ).
$r$$3^{n}$$r_{0},$$r$$r_{1}\ldots$

Variation 2. Just as Young’s lattice is associated with the ordinary representations of, so the shifted Young’s lattice is associated with the projective representations of .
is defined to be the subposet (actually a sublattice) of consisting of all partitions with( is unchanged)
distinct parts. By a suitable modification of the linear transformation$=I$
relationallowswe still have the fundamental
DU–UD“differential” proofs of. This
well-known formulae and some new generalizations of them concerning shifted tableaux. The$S_{n}$$\tilde{Y}$$S_{n}$$Y$$\tilde{Y}$$U$$\neq$$D$

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most well-known of these formulae is$\sum_{\mu}2^{n-\ell(\mu)}(g^{\mu})^{2}=n!$,is the length of , andwhere ranges over all partitions of into distinct parts, whereis the number of standard shifted tableaux of shape (i.e., the number of saturated
wherechains in from to ).
$\ell(\mu)$

$n$

$\mu$

$g^{\mu}$$\mu$$\mu$$\tilde{Y}$$\emptyset$$\mu$

For further information on generalizations and extensions of differential posets, see [2].

6. Open problems. We mentioned in Section 2 the problem of characterizing differen. we mention
tial lattices, and of finding a “nice” combinatorial description of the lattices
one further open problem here; more can be found in Section 6 of [1].
$\mathcal{F}_{n}$Problem. Fix a positive integer . What is the greatest (respectively, least) number ofelements of rank that an r-differential poset can have? It seems plausible that the extreme, respectively. Alongvalues are achieved by $Z(r)$ (the r-differential Fibonacci lattice) andthe same lines, given that $Pj=\# P_{j}$ for some , what is the largest (respectively, smallest)? Do we always have $p_{j+1}\leq rp_{j}+p_{j-1}$ ? Do we always have $p_{J+1}>p_{j}$ ,cardinality of)
except when $r=1$ and $j=0$ ? (It’s easy to see that we always have
$r$$n$$Y^{r}$$j$$P_{j+1}$$p_{J+1}\geq p_{j}.$REFERENCES

1. R. Stanley, Differential posets, J. Amer.

$A\eta fath$. Soc. 1 (1988), 919-961.

2. R. Stanley, Variations on differential posets, in Invariant Theory and Tableaux (D.
Stanton, ed.), The IMA Volumes in Mathematics and Its Applications, vol. 19, SpringerVerlag, New York, 1990, pp. 145-165.

3. R. Stanley, Further combinatorial properties of two Fibonacci lattices, European J.
Combinatorics 11 (1990), 181-1SS.

$\zeta$