2003/03/09 by Dmitri I. Panyushev
Mathematics · #math.RT #math.CO
published as J. Algebra 274 (2004), 822-846 · LaTeX2e, 23 pages
arxiv created 2003/03/09 · arxiv updated 2009/11/30
It was shown by Cellini and Papi that an ad-nilpotent ideal determines certain element of the affine Weyl group, and that there is a bijection between the ad-nilpotent ideals and the integral points of a simplex with rational vertices. We give a description of the generators of ad-nilpotent ideals in terms of these elements, and show that an ideal has k generators if and only it lies on the face of this simplex of codimension k. We also consider two combinatorial statistics on the set of ad-nilpotent ideals: the number of simple roots in the ideal and the number of generators. Considering the first statistic reveals some relations with the theory of clusters (Fomin-Zelevinsky). The distribution of the second statistic suggests that there should exist a natural involution (duality) on the set of ad-nilpotent ideals. Such an involution is constructed for the series A,B,C.