Minuscule elements of Weyl groups, the numbers game, and \(d\)-complete posets. (English) Zbl 0969.05068

Proctor, Robert A.

J. Algebra 213, No. 1, 272-303 (1999).

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

Keywords:

numbers game; distributive lattices; join irreducibles; \(d\)-complete posets; integrable representations of simply laced Kac-Moody algebras; intervals of weak Bruhat orders; Bruhat lattices; Weyl groups; Schubert varieties

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