Minuscule elements of Weyl groups.

*(English)*Zbl 0973.17034From the beginning of R. A. Proctor’s work [J. Algebra 213, 272-303 (1999; Zbl 0969.05068)], it has been clear that \(\lambda\)-minuscule elements are “fully commutative” in the sense of J. R. Stembridge [J. Algebr. Comb. 5, 353-385 (1996; Zbl 0864.20025)]. The first objective of the paper under review is to clarify more directly the exact nature of the relationship, providing reduced-word characterizations of minuscule elements as well as order-theoretic characterizations of their “heaps”.

The second objective is to extend Proctor’s classification of (dominant) \(\lambda\)-minuscule elements (or equivalently, their heaps) from the simply-laced case to any symmetrizable Kac-Moody Weyl group. There is a natural way to decompose heaps of dominant minuscule elements into irreducible components. In the simply-laced case, R. A. Proctor has shown that the irreducible cases can be grouped into 15 families, 14 of which are infinite [J. Algebr. Comb. 9, 61-94 (1999; Zbl 0920.06003)]. In the present paper, the author shows that in the multiply-laced cases, there are two more infinite families.

In the final section, the author shows that the heap of any (dominant) \(\lambda\)-minuscule element \(w\) can also be obtained by restricting the standard partial ordering of the positive co-roots to those co-roots that are “inverted” by \(w\) (i.e., \(\alpha\check{}>0\) and \(w\alpha\check{}<0\)).

The second objective is to extend Proctor’s classification of (dominant) \(\lambda\)-minuscule elements (or equivalently, their heaps) from the simply-laced case to any symmetrizable Kac-Moody Weyl group. There is a natural way to decompose heaps of dominant minuscule elements into irreducible components. In the simply-laced case, R. A. Proctor has shown that the irreducible cases can be grouped into 15 families, 14 of which are infinite [J. Algebr. Comb. 9, 61-94 (1999; Zbl 0920.06003)]. In the present paper, the author shows that in the multiply-laced cases, there are two more infinite families.

In the final section, the author shows that the heap of any (dominant) \(\lambda\)-minuscule element \(w\) can also be obtained by restricting the standard partial ordering of the positive co-roots to those co-roots that are “inverted” by \(w\) (i.e., \(\alpha\check{}>0\) and \(w\alpha\check{}<0\)).

Reviewer: Vladimir L. Popov (Moskva)

##### MSC:

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

06A07 | Combinatorics of partially ordered sets |

05E10 | Combinatorial aspects of representation theory |

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\textit{J. R. Stembridge}, J. Algebra 235, No. 2, 722--743 (2001; Zbl 0973.17034)

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##### References:

[1] | Bourbaki, N., Chaps. 4-6, Groupes et algèbres de Lie, (1981), Masson Paris |

[2] | C. K. Fan, A Hecke Algebra Quotient and Properties of Commutative Elements of a Weyl Group, Ph. D. thesis, Massachusetts Institute of Technology, 1995. |

[3] | Fan, C.K.; Stembridge, J.R., Nilpotent orbits and commutative elements, J. algebra, 196, 490-498, (1997) · Zbl 0915.20019 |

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[6] | D. Peterson, and, R. A. Proctor, in preparation. |

[7] | Proctor, R.A., Minuscule elements of Weyl groups, the numbers game, and d-complete posets, J. algebra, 213, 272-303, (1999) · Zbl 0969.05068 |

[8] | Proctor, R.A., Dynkin diagram classification of λ-minuscule Bruhat lattices and of d-complete posets, J. algebraic combin., 9, 61-94, (1999) · Zbl 0920.06003 |

[9] | R. A. Proctor, Generalized Young diagrams with well-defined jeu de taquin emptying procedures, preprint. |

[10] | Stembridge, J.R., On the fully commutative elements of Coxeter groups, J. algebraic combin., 5, 353-385, (1996) · Zbl 0864.20025 |

[11] | J. R. Stembridge, Quasi-minuscule quotients and reduced words for reflections, J. Algebraic Combin. to appear. · Zbl 0981.20030 |

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