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Dynkin diagram classification of \(\lambda\)-minuscule Bruhat lattices and of \(d\)-complete posets. (English) Zbl 0920.06003
Summary: \(d\)-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.

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)
Full Text: DOI
[1] Alon, N.; Krasikov, I.; Peres, Y., Reflection sequences, American Math. Monthly, 96, 820-822, (1989) · Zbl 0714.20023
[2] A. Björner, “On a combinatorial game of S. Mozes,” preprint, 1988.
[3] K. Eriksson, “Strongly convergent games and Coxeter groups,” Ph.D. Thesis, Kungl Tekniska Högskolan, 1993.
[4] E. Gansner, “Matrix correspondences and the enumeration of plane partitions,” Ph.D. Thesis, M.I.T., 1978.
[5] Lakshmibai, V., Bases for Demazure modules for symmetrizable Kac-Moody algebras, Linear Algebraic Groups and Their Representations, 153, 59-78, (1993) · Zbl 0807.17018
[6] Mozes, S., Reflection processes on graphs and Weyl groups, J. Combinatorial Theory, 53, 128-142, (1990) · Zbl 0741.05035
[7] R. Proctor, “Minuscule elements of Weyl groups, the numbers game, and \(d\)-complete posets,” J. Algebra, to appear. · Zbl 0969.05068
[8] Proctor, R., Bruhat lattices, plane partition generating functions, and minuscule representations, European J. Combinatorics, 5, 331-350, (1984) · Zbl 0562.05003
[9] Proctor, R., A Dynkin diagram classification theorem arising from a combinatorial problem, Advances Math, 62, 103-117, (1986) · Zbl 0639.06006
[10] R. Proctor, “Poset partitions and minuscule representations: External construction of Lie representations, Part I,” preliminary manuscript.
[11] Sagan, B., Enumeration of partitions with hooklengths, European J. Combinatorics, 3, 85-94, (1982) · Zbl 0483.05010
[12] R. Stanley, “Ordered structures and partitions,” Memoirs of the AMS119 (1972). · Zbl 0246.05007
[13] R. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth & Brooks/Cole, Monterey, 1986. · Zbl 0608.05001
[14] Stembridge, J., On the fully commutative elements of Coxeter groups, J. Alg. Combin, 5, 353-385, (1996) · Zbl 0864.20025
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