Dynkin diagram classification of \(\lambda\)-minuscule Bruhat lattices and of \(d\)-complete posets.

*(English)*Zbl 0920.06003Summary: \(d\)-complete posets are defined to be posets which satisfy certain local structural conditions. These posets play or conjecturally play several roles in algebraic combinatorics related to the notions of shapes, shifted shapes, plane partitions, and hook length posets. They also play several roles in Lie theory and algebraic geometry related to \(\lambda\)-minuscule elements and Bruhat distributive lattices for simply laced general Weyl or Coxeter groups, and to \(\lambda\)-minuscule Schubert varieties. This paper presents a classification of \(d\)-complete posets which is indexed by Dynkin diagrams.

##### MSC:

06A07 | Combinatorics of partially ordered sets |

05E10 | Combinatorial aspects of representation theory |

17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |

20F55 | Reflection and Coxeter groups (group-theoretic aspects) |

##### Keywords:

\(d\)-complete posets; \(\lambda\)-minuscule elements; \(\lambda\)-minuscule Schubert varieties; minuscule Weyl group element; reduced decomposition; simply laced Weyl group; Bruhat order; algebraic combinatorics; shifted shapes; plane partitions; hook length posets; Lie theory; algebraic geometry; Bruhat distributive lattices; Coxeter groups; Dynkin diagrams
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\textit{R. A. Proctor}, J. Algebr. Comb. 9, No. 1, 61--94 (1999; Zbl 0920.06003)

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