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Minuscule elements of Weyl groups, the numbers game, and $$d$$-complete posets. (English) Zbl 0969.05068
Summary: 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 of $$d$$-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 $$\lambda$$-minuscule elements of Weyl groups and their associated Schubert varieties. The $$d$$-complete posets have recently been proven to possess both the hook length and the jeu de taquin properties.

##### MSC:
 05E15 Combinatorial aspects of groups and algebras (MSC2010) 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 06A06 Partial orders, general 20G05 Representation theory for linear algebraic groups
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