Weyl Groups, the Hard Lefschetz Theorem, and the Sperner Property

Techniques from algebraic geometry, in particular the hard Lefschetz theorem, are used to show that certain finite partially ordered sets O x derived from a class of algebraic varieties X have the k-Sperner property for all k. This in effect means that there is a simple description of the cardinality of the largest subset of C) x containing no (k + 1)-element chain. We analyze, in some detail, the case when X G/P, where G is a complex semisimple algebraic group and P is a parabolic subgroup. In this case, Qx is defined in terms of the "Bruhat order " of the Weyl group of G. In particular, taking P to be a certain maximal parabolic subgroup of G SO(2n + 1), we deduce the following conjecture of Erd6s and Moser: Let S be a set of 2 + 1 distinct real numbers, and let T1, , Tk be subsets of S whose element sums are all equal. Then k does not exceed the middle coefficient of the polynomial 2(1 + q)2(1 + q2)2... (1 + qe)2, and this bound is best possible. 1. The Sperner property. Let P be a finite partially ordered set (or poser, for short), and assume that every maximal chain of P has length n. We say that P is graded of rank n. Thus P has a unique rank function p:P- {0, 1,..., n} satisfying p(x) = 0 if x is a minimal element of P, and p(y) p(x) + 1 if y covers x in P (i.e., if y> x and no z 6 P