Short antichains in root systems, semi-Catalan arrangements, and B-stable subspaces

Abstract

Let G be a complex simple algebraic group with Lie algebra g. Fix a Borel subalgebra b. An ideal of b is called ad-nilpotent, if it is contained in [b, b]. The goal of this paper is to present a refinement of the enumerative theory of ad-nilpotent ideals in the case, where g has roots of different length. Let Ad denote the set of all ad-nilpotent ideals of b. Any c ∈ Ad is completely determined by the corresponding set of roots. The minimal roots in this set are called the generators of an ideal. The collection of generators of an ideal forms an antichain in the poset of positive roots, and the whole theory can be expressed in the combinatorial language, in terms of antichains. An antichain is called strictly positive, if it contains no simple roots. Enumerative results for all and strictly positive antichains were recently obtained in the work of Athanasiadis, Cellini–Papi, Sommers, and this author [1–4, 9, 13]. There are two different theoretical approaches to describing (enumerating) antichains. The first approach consists of constructing a bijection between antichains and the coroot lattice points lying in a certain simplex. An important intermediate step here is a bijection between antichains and the so-called minimal elements of the affine Weyl group, Ŵ . It turns out that the simplex obtained is “equivalent” to a dilation of the fundamental alcove of Ŵ , so that the problem of counting the coroot lattice points in it can be resolved. For strictly positive antichains, one constructs another bijection and another simplex, and the respective elements of Ŵ are called maximal; yet, everything is quite similar. The second approach uses the Shi bijection between the ad-nilpotent ideals (or antichains) and the dominant regions of the Catalan arrangement. Under this bijection, the strictly positive antichains correspond to the bounded regions. There is a powerful result of Zaslavsky allowing one to compute the number of all and bounded regions, if the characteristic polynomial of the arrangement is known. Since the characteristic polynomial of the Catalan arrangement was recently computed in [1], the result follows. If g has roots of different length, one can distinguish the length of elements occurring in antichains. We say that an antichain is short, if it consists of only short roots. This notion has a natural representation-theoretic incarnation: the short antichains are in