ad-Nilpotent ideals of a Borel subalgebra

Cellini, Paola; Papi, Paolo

doi:10.1006/jabr.1999.8099

Cited by 82 publications

(181 citation statements)

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“…In his study of sign types for affine Weyl groups [20,21], Shi also showed that the set of ideals in Φ + is in bijection with the set of positive sign types for W and with the set of regions into which the fundamental chamber is dissected by the hyperplanes of Cat Φ . A bijection between ideals in Φ + and W -orbits of T was later described by Cellini and Papi [5] and is based on the construction of a map which assigns an element of the affine Weyl group W a to each ideal in Φ + [4]. In the special case m = 1, the expression (1.1) is refered to as the Catalan number associated to Φ [2,17], since it reduces to the familiar n th Catalan number for the root system A n−1 .…”

Section: Introduction and Resultsmentioning

confidence: 99%

On a refinement of the generalized Catalan numbers for Weyl groups

Athanasiadis

2004

Trans. Amer. Math. Soc.

127

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Abstract. Let Φ be an irreducible crystallographic root system with Weyl group W , coroot latticeQ and Coxeter number h, spanning a Euclidean space V , and let m be a positive integer. It is known that the set of regions into which the fundamental chamber of W is dissected by the hyperplanes in V of the form (α, x) = k for α ∈ Φ and k = 1, 2, . . . , m is equinumerous to the set of orbits of the action of W on the quotientQ/ (mh + 1)Q. A bijection between these two sets, as well as a bijection to the set of certain chains of order ideals in the root poset of Φ, are described and are shown to preserve certain natural statistics on these sets. The number of elements of these sets and their corresponding refinements generalize the classical Catalan and Narayana numbers, which occur in the special case m = 1 and Φ = A n−1 .

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Section: Introduction and Resultsmentioning

confidence: 99%

On a refinement of the generalized Catalan numbers for Weyl groups

Athanasiadis

2004

Trans. Amer. Math. Soc.

127

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“…Moreover, we say that V is principal if the corresponding ideal is: this amounts to say that V has a minimum. A useful technique for dealing with the abelian or with the ad-nilpotent ideals of a Borel subalgebra is to see them as special subsets of the affine root system associated to Φ (see [6]). The idea, for the abelian ideals, is due to D. Peterson, and was first described in [15].…”

Section: Conversely Each Ideal Of B Included Inmentioning

confidence: 99%

Root Polytopes and Borel Subalgebras

Cellini¹,

Marietti

2014

International Mathematics Research Notices

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“…The general theory of abelian ideals (of b) is based on a relationship with the so-called minuscule elements of the affine Weyl group W (the Kostant-Peterson theory, see [8] and also [3]). Proofs in this article heavily rely on some further results obtained in [10,11].…”

Section: Definition 1 ([9]) a Subset F Of W Is A Minimal Inversion Cmentioning

confidence: 99%