Minuscule elements of Weyl groups. (English) Zbl 0973.17034

From 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 \(w\alpha\check{}<0\)).


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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