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

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

2003-04-24

arxiv.org/abs/math/0304380v1

Europ. J. Combinatorics, 25 (2004), 93--112

Mathematics

Combinatorics

LaTeX2e, 20 pages

Scientific paper

Let $\be$ be a Borel subalgebra of a complex simple Lie algebra $\g$. An ideal of $\be$ is called ad-nilpotent, if it is contained in $[\be,\be]$. The generators of an ad-nilpotent ideal give rise to an antichain in the poset of positive roots, and the whole theory can be expressed in a combinatorial fashion, in terms of antichains. The aim of this paper is to present a refinement of the enumerative theory of ad-nilpotent ideals for the case in which $\g$ has roots of different length. An antichain is called short, if it consists of short roots. We obtain, for short antichains, analogues of all results known for the usual antichains.

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