Proceedings of the International Congress of Angust 16-24, 1983, Warszawa

Mathematicians

RICHARD P. STANLEY*

Combinatorial Applications of the Hard Lefschetz Theorem

1. The hard Lefschetz theorem

Let X be a smooth irreducible complex projective variety of (Complex) diiliellsion m (or more generally a KBhlcr manifold), endowed with the "clas- sical" Hausdorff topology. Let H* (X) = E" ( X ) @HI (X) @ . . . O F n (3) clenote the singuler cohomology ring of X with complex coefficients. (Any field of characteristic zero would do just a8s well for the coefficient group. In fact', for the most part we could work over 2, but this is unneces- sary for our purposes.) Since X is'projective we may imbed it in some complex projective space pN. Letl H denote a, (generic) hyperplane in PN. Thcn X n H is a closed subvariety of X of real codimension two, and thus by a, standard construction in algebraic geometry represents a cohomology class w E H2(X), called the class of a Ayperplane section.

T m HARD LEFSCHETZ THXOREM. Let 0 < i ,< m. The map H ~ ( x ) ---t

azn-i (X) given by muliiplication by con-$ is am isomorphism of vector spaces.

This result wss first stated by Lefsclietz in [IS], but his proof was not entirely rigorous. The first complete proof was given by Hodge in [15], using his theory of harmonic integrals. The "standard" proof today uses the representation theory of the Lie algebra sl,(C) and is due to Chern [4]. Lefschetz' original proof was only recently made rigorous by Deligne (see [92 ] ) , who extended it to characteristic p. Other references include t51, [131, [171, 1321.

* Parhiz~lly supported by the National Science Foundation.

Section 2: R. P. Stanley

2. Unimodality

Since wn-5 HHf(X)-+E2n-i(X) is bijective for all 0 < i < n, it follows that w : Hi(X)-+H'+' (X) is iqzjective for 0 < i < n -1 and surjective for 3~ < i ,( 2%-1. Thus if Pi = &(X) = d ima i (X) denot3es the i-t'h Betti number of X, then tho sequences Po, P,, . . . , /Ipn and PI, Ps, . . ., pzn-' are symmi~lric and unimodal, i.e., p, < ,8, ,< . . . < @2~,,,1 and Pi = Pzn-i, and similarly for PI, pa, . . . , pZn-, . This consequence of the hard Lefschetz thcorm was well known from the beginning.

There are several examples for which p, has a combinatorial intcrprc- tation. The archetype is the Grassmann vaziety X = G,,, of d-planes in Cn. It was rigorously known since Ehresmann [ti] tha t /IBli+' = 0 and that P2$ is the number p ( i , d, n-d) of partitions of the integer i into < d parts, with largest part < 9% -d. The unimodality of the sequence p (0 , d, n-d), p (1 , d ,n -d ) , ..., p(d(n-d) , d, n-d) was first proved by Sylvester [31], and several subsequent proofs have been given. Perhaps the simplest is [28, Cor. 9.61, but no purely combinatorial proof is known. Such a proof would.involve an explicit injection from the partitions counted b y p ( i , d ,n -d ) to thosecounted b y p ( i f 1 , d ,n-@,for O < i < [ 3 d ( n - -41.

More generally, take X to be a generalized flag manifold G/P, where G is a complex semisimple algebraic group and P a parabolic subgroup. The hard Lefschetz theorem then yields the uniniodality of the numbcr of elements of lcngth i in the quotient Bruhat order WJ = W/WJ7 where W is the Weyl group of G and W j cf P ( [26, fj 31).

Now let g be a complex semisimple Lie algebra and L, an irreducible finite-dinlcnsionsl g-module with highest weight A. Let m,(p) denote the multiplicity of the weight p, in L,. The height h t p of p is tho number of simple roots which need to be added to - -A to obtain p. The polynomial PA(¶) = 2 m,(p)$tp mas first shown by Dynkin (see [7, p. 3321) to have

C

unimodal coefficients; see also [25]. Rccently Lusztig [I91 has conq~uted the (complex) intersection cohomology H*(8,), as defined by Goresky and ISlacpherson [lo], [Ill, of certain Schz~bert varieties 0, (in gcneral sin- gular). Namely,

2 dim ~ ~ ( 8 , ) pi = P,(p2). i

(See [19, Cor. 8.91). Now the intersection cohomology I i * ( X ) of any complex projective variety, considered as a module over the singular cohomology

Combinatorial Applications of tho Herd Lefschetz Theorem

H * ( X ) , satisfies the hard Lefschetz theorem. Hence Dynkin's result may be regarded as a consequence of the hard Lefschetz theorem for inter- section cohomology. It would be interesting to investigah what other sequences of combinatorial interest arise as Betti numbers in intersection cohomology . 3. McMullen's g-conjecture

Let 9 be a d-dimensional simplical convex polytope ([14], 1211) with fi i-dimensional faces, 0 < i < d-1. We call the vector f (9) = (f,, . . . , f,-,) the f-vector of P. The problem of obtaining information about such vectors goes back to Descartes and Euler. In 1971 McMullen [20], [21, p. 1791, drawing on all the available evidence, gave s remarkable condition on a vector (f,, . . . , fd-,) which he conjectured mas equivalent to being an f-vector as above.

To describe this condition, define a new vector h(9) = (h,, ..., h,), called the h-vector of 9, by

where wo set f-, = 1. The Dehn-Sommerville equations ([14, ss9.2, 9-81, [21, ss2.4, 5.11) assert that hi = h,-,. McMullen's conditions, though ho did not realize it, are equivalent to the Dehn-Sommerville equations together with the existence of a graded commutative C-algebra R = R O B OR1@. . . where R, = C, R is generated as a C-algebra by R,, and dim, R, = hi - hi-, for 1 ,( i < [d/2]. In particular, h, < h, < . . . Q h~dl,l, and we arc led to suspect the existence of a smooth d-dimensional complex pro- jective variety X ( 9 ) for which PZi ( X ( 9 ) ) = hi. I f moreover H* (X(9)) is generated by H2(X(9)), then we can take R = H*(X(9))/(m), where (o) denotes the ideal generated by the class of a hyperplane section, to deduce the necessity of ~ c ~ u l l e n ' s conditions. In [6] varieties X ( 9 ) arc constructed (after some assumptions on 9 irrelevant for proving Mc- Mullen's conjecture) with all the desired properties except smoothness (despite the misleading statement in [6, Rmk. 3-81). Although X ( 9 ) need not be smooth, its singularities are sufficiently nice that the hard Lefsclletz theorem continues to hold [30]. Namely, X ( 9 ) is a V-variety, i.e., locnlly it looks like Cn/G where G is a finite group of linear trxnsfor- mations. Thus the necessity of McMullen's condition follows [27]. Suffi- ciency was proved about tho same time by Billers and Lee [I], [2]. For

Section 2: R. P. Stanley

further information see [29]. Some recent work of Ealai [16] suggests that a more elementary proof may be possible, and perhaps an extension to more general objects (such as shellable triangulations of spheres or even triangulations of homology spheres).

4. The Sperner property

Let P be s finite poset ( = partially ordered set). We say P is graded of rank n if every maximal chain of P has length n (or cardinality n+l) . TVe then define the rank q(x) of x e P to be the length 1 of the longest chainx,<x,< ... <al =x. Let Pi = {xEP: ~ ( x ) =i). An mtichain is a subset A of P of pairwise incomparable elements. Thus each P, is an antichain. We say that P has the Sperner property if no antichain is larger than the largest Pi. This terminology stems from the theorem of E. Sperner [24] that the poset of subsets of an %-element set, ordered by inclusion, has t'he Sperner property (see also [12]).

I f now X is a complex projective variety, we say X has a cellular de- composition if there exists a (finite) set V = {C,, . . . , 0,) of pairwise disjoint subsets Ci of X, each isomorphic as algebraic varieties to complex affine space Cn$, such that u C, = X and the closure Ei (in the classical or Zariski topology) of each Ci is a union of 0,'s. We then define a poset QX = QX(%') to be the set V ordered by reverse inclusion of the closure of the C,'s. If X is irreducible of dimension n, then Qx is graded of rank n.

THEOREX 1 ([26, Thm. 2.41). If X (os above) is smooth and irreducibZe, then QX has the Bperner p~operty.

The crucial step in the proof is the use of the hard Lefschetz theorem, and indeed the theorem remains t'rue for any irreducible X (with a cellular decomposition) satisfying the conclusions of the hard Lefschetz theorem. The conclusion to the above theorem is not the strongest possible; see 1261 for further details.

The main class of varieties to which the above theorem applies (indeed, the only known class for which the Sperner property of QX is non-trivial) are $he generalized flag manifolds X = G/P mentioned above. Here the QX &re the quotient Bruhat orders WJ. For certain choices of 8 and P thc posets QX have a, special combinatorial significance. I n particular, taking@ = SO (2n $1, Cjand a cei-tain maximal P, the Spernicity of the poset QX can be used to prove the following number-theoretic conjecture of Erdiis and Maser [Q, eqn. (12)l.

Combinatorial Applications of the Herd Lefschetz Theorem

THEOREM 2. Let 8 be a finite subset of R, and for k E R let f (8,k) clenots the .number of subsets of 8 whose elements sum to k. Then for 181 = 21 +1, we have

For further information in addition to [26], see [23] and the references therein.

I t would be interesting to discover other properties of the posets QX of Theorem 1 (in addition to the Sperner property). In particular, if A denotes the simplicia1 complex of chains of OX, where QX denotes Q" with the bottom and top element removed, then is the geometric, reali- zation 1 A1 always a sphere or cell4 (It may not even be necessary to assume X-'is smooth.) This is true when dimX < 2 or when X = G/P. For the latter case see [3].

In [25, Problem 21 it is asked whether some posets arising from irredu- cible representations of the Lie algebra sl(.n, C) have a "symmetric chain decomposition", which is stronger than the Sperner property. I n fact, even the Sperner property is open. Perhaps there is a "cellular decompo- sition for intersection cohomology" of tho Schubert varieties 8, discussed in Section 2 which would yield a proof of the Sperner property.

References

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[3] Bjorner A. and Wachs M., Bruhat Order of Coxeter Groups and Shellability, Advances i n Math. 43 (1982), pp. 87-100.

[4] Chern S.-S., On a Generalization of Kiihler Geometry. In : R. H. Pox, et al . (eds.), Algebraic Geometry and Topology, Princeton University Press, Princeton, N. J., 1951, pp. 103-124.

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Section 2: R. P. Stanley

[8] Ehresmann C., Sur la topologie de certains espaces homog&nes, Ann. Bath. 35 (1934), pp. 396-443.

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1231 Proctor R., Representations of sl(2, C ) on Posets and tho Sperner Property, SIAM J . Alg. Disc. Math. 3 (1982), pp. 276-280.

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Combinatorial Applications of the Hard Lefsohetz Theorem

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