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

Abstract

Techniques from algebraic geometry, in particular the hard Lefschetz theorem, are used to show that certain finite partially ordered sets $Q^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 $Q^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, $Q^X $ 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 Erdös and Moser: Let S be a set of $2\ell + 1$ distinct real numbers, and let $T_1 , \cdots ,T_k $ 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 + q^2 )^2 \cdots ( 1 + q^\ell )^2 $, and this bound is best possible.