Dynkin Diagram Classification of λ -Minuscule Bruhat Lattices and of d-Complete Posets
Robert A. Proctor
Department of Mathematics University of North Carolina Chapel Hill North Carolina 27599
DOI: 10.1023/A:1018615115006
Abstract
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 
-minuscule elements and Bruhat distributive lattices for simply laced general Weyl or Coxeter groups, and to -minuscule Schubert varieties. This paper presents a classification of d-complete posets which is indexed by Dynkin diagrams.
-minuscule elements and Bruhat distributive lattices for simply laced general Weyl or Coxeter groups, and to -minuscule Schubert varieties. This paper presents a classification of d-complete posets which is indexed by Dynkin diagrams.Pages: 61–94
Keywords: d-complete poset; minuscule Weyl group element; reduced decomposition; Dynkin diagram
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References
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2. A. Bj\ddot orner, “On a combinatorial game of S. Mozes,” preprint, 1988.
3. K. Eriksson, “Strongly convergent games and Coxeter groups,” Ph.D. Thesis, Kungl Tekniska H\ddot ogskolan, 1993.
4. E. Gansner, “Matrix correspondences and the enumeration of plane partitions,” Ph.D. Thesis, M.I.T., 1978.
5. V. Lakshmibai, “Bases for Demazure modules for symmetrizable Kac-Moody algebras,” in Linear Algebraic Groups and Their Representations, pp. 59-78, Contemporary Math. 153, AMS, Providence, 1993.
6. S. Mozes, “Reflection processes on graphs and Weyl groups,” J. Combinatorial Theory 53 (1990), 128-142.
7. R. Proctor, “Minuscule elements of Weyl groups, the numbers game, and d-complete posets,” J. Algebra, to appear.
8. R. Proctor, “Bruhat lattices, plane partition generating functions, and minuscule representations,” European J. Combinatorics 5 (1984), 331-350.
9. R. Proctor, “A Dynkin diagram classification theorem arising from a combinatorial problem,” Advances Math. 62 (1986), 103-117.
10. R. Proctor, “Poset partitions and minuscule representations: External construction of Lie representations, Part I,” preliminary manuscript.
11. B. Sagan, “Enumeration of partitions with hooklengths,” European J. Combinatorics 3 (1982), 85-94.
12. R. Stanley, “Ordered structures and partitions,” Memoirs of the AMS 119 (1972).
13. R. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth & Brooks/Cole, Monterey, 1986.
14. J. Stembridge, “On the fully commutative elements of Coxeter groups,” J. Alg. Combin. 5 (1996), 353-385.