ad-nilpotent ideals of a Borel subalgebra: generators and duality

By Dmitri I. Panyushev. 2003

55 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b5783f9472070df527346f41be9f5d2a33ab1275

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Citations in the corpus

On a refinement of the generalized Catalan numbers for Weyl groups (2004, unknown): “Lemma 2.1. (i) If 1;2;:::;r2 + with r 2 and = 1 +2+ +r2 +, then there exists i with 1 i r such that i2 +. (ii) ([ 14 , 22]) If 1;2;:::;r2 and 1 +2 + +r = 2 ,t hen1 = or there exists i with 2 i r such that 1 +i2 [f0g.”
B-Stable Ideals in the Nilradical of a Borel Subalgebra (2003, result): “Finally, we note that Athanasiadis has obtained our second theorem (by a different method) [1], as has Panyushev, who has also obtained some of our results on minimal roots [11].”
Tilting theory and cluster combinatorics (2004, background): “... of the dual canonical basis of the quantised enveloping algebra of a semisimple Lie algebra over C, and that it should model the (classical and quantised) coordinate rings of varieties associated to algebraic groups (now shown in several cases — see [BFZ02], [Sco03].), with particular relevance ”
Root systems and generalized associahedra (2005, background): “9 is a combination of results in [3, 19, 39, 44, 48]; see [3] for a historical overview, and for further generalizations.”