Proceedings of the International Congress of Mathematicians
Angust 16-24, 1983, Warszawa
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 "classical" 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 a8swell for the coefficient
group. In fact', for the most part we could work over 2, but this is unnecessary 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 [ 9 2 ] ) , who extended it to characteristic p. Other references include
t51, [131, [171, 1321.
*
Parhiz~llysupported 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 i m a i ( X ) denot3es t h e i-t'h Betti number
of X, then tho sequences Po, P,, ...,/Ipnand 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 t h e hard Lefschetz thcorm was
well known from t h e beginning.
There are several examples for which p, has a combinatorial intcrprctation. The archetype is the Grassmann vaziety X = G,,, of d-planes in
Cn. It was rigorously known since Ehresmann [ti] t h a t /IBli+' = 0 and
t h a t P2$is t h e number p ( i , d, n-d) of partitions of the integer i into
< d parts, with largest part < 9%-d. The unimodality of the sequence
p ( 0 , d, n-d), p ( 1 , d , n - d ) , ...,p ( d ( n - d ) , d, n-d) was first proved by
Sylvester [31], and several subsequent proofs have been given. Perhaps
the simplest is [28, Cor. 9.61, but no purely combinatorial proof is known.
Such a proof would.involve an explicit injection from t h e partitions counted
b y p ( i , d , n - d ) t o thosecounted b y p ( i f 1 , d , n - @ , f o r O < i < [ 3 d ( n -
...
.
..
.
-41.
More generally, take X to be a generalized flag manifold G/P, where
G is a complex semisimple algebraic group and P a parabolic subgroup.
The hard Lefschetz theorem then yields t h e uniniodality of the numbcr
of elements of lcngth i in the quotient Bruhat order WJ = W/WJ7 where
W is t h e Weyl group of G and W j cf P ( [26, fj31).
Now let g be a complex semisimple Lie algebra and L, a n irreducible
finite-dinlcnsionsl g-module with highest weight A. Let m,(p) denote the
multiplicity of t h e weight p, in L,. The height h t p of p is tho number of
simple roots which need t o be added t o --A to obtain p. The polynomial
PA(¶)
=
m,(p)$tp mas first shown by Dynkin (see [7, p. 3321) to have
2
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~bertvarieties 0, (in gcneral singular). Namely,
2 dim ~ ~ ( pi8 , )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 t h e 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 intersection 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-dimensionalfaces, 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]. I n particular, h, < h, < . Q h~dl,l,and we
arc led to suspect the existence of a smooth d-dimensional complex projective 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, t o
deduce the necessity of ~ c ~ u l l e nconditions.
's
In [6] varieties X ( 9 ) arc
constructed (after some assumptions on 9 irrelevant for proving McMullen'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 trxnsformations. Thus the necessity of McMullen's condition follows [27]. Sufficiency 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
c h a i n x , < x , < ... < a l =x. Let Pi = { x E P : ~ ( 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 de0,) of pairwise disjoint
composition if there exists a (finite) set V = {C,,
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.
...,
THEOREX1([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 Q X
of Theorem 1 (in addition to the Sperner property). I n 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, realization 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].
I n [25, Problem 21 it is asked whether some posets arising from irreducible 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 decomposition for intersection cohomology" of tho Schubert varieties 8, discussed
in Section 2 which would yield a proof of the Sperner property.
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33
- Proceedings...
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