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 satisfies y > z > x). If p (x) i, then we say that x has rank i. Define Pi {x P: p (x) i} and set pi pi(P) card Pi. The polynomial F(P, q) po + plq +" + Pnq is called the rank-generating function of P. We say that P is rank-symmetric if pi pn-for all i, and that P is rank-unimodal if po <= pl <=" <= pi >= p+ >=" >= pn for some i. An antichain (also called a Spernerfamily or clutter) is a subset A of P, such that.no two distinct elements of A are comparable. The poset P is said to have the Sperner property (or property $1) if the largest size of an antichain is equal to max {pi: 0 <= <-n}. More generally, if k is a positive integer then P …