Minuscule elements of Weyl groups, the numbers game, and d-complete posets

Certain posets associated to a restricted version of the numbers game of Mo tes are shown to be distributive lattices. The posets of join irreducibles of these distributive lattices are characterized by a collection of local s tructural properties, which form the definition of d-complete poser. In rep resentation theoretic language, the top "minuscule portions" of weight diag rams for integrable representations of simply laced Kac-Moody algebras are shown to be distributive lattices. These lattices form a certain family of intervals of weak Bruhat orders. These Bruhat lattices are useful in studyi ng reduced decompositions of lambda-minuscule elements of Weyl groups and t heir associated Schubert varieties. The d-complete posets have recently bee n proven to possess both the hook length and the jeu de taquin properties. (C) 1999 Academic Press.