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Ad-nilpotent ideals of a Borel subalgebra: generators and duality
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 generators if and only
it lies on the face of this simplex of codimension . 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.Comment: LaTeX2e, 23 page
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