Minuscule Elements of Weyl Groups, the Numbers Game, andd-Complete Posets

Certain posets associated to a restricted version of the numbers game of Mozes are shown to be distributive lattices. The posets of join irreducibles of these distributive lattices are characterized by a collection of local structural properties, which form the definition ofd-complete poset. In representation theoretic language, the top “minuscule portions” of weight diagrams 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 studying reduced decompositions of λ-minuscule elements of Weyl groups and their associated Schubert varieties. Thed-complete posets have recently been proven to possess both the hook length and the jeu de taquin properties.