Ad-nilpotent ideals of a Borel subalgebra II

Cellini, Paola; Papi, Paolo

doi:10.1016/s0021-8693(02)00532-x

Cited by 75 publications

(89 citation statements)

References 14 publications

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“…These Catalan and Narayana numbers are generalized by the numbers which appear in Theorems 1.1 and 1.2, respectively. In the case m = 1, it turns out that the composite of our two bijections coincides with the one constructed in [5]. However, our motivation for constructing the second bijection comes from the paper by Shi [20], in which an element of the affine Weyl group is associated to each sign type of Φ in order to count the total number of sign types.…”

Section: Introduction and Resultsmentioning

confidence: 74%

“…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: 96%

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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: 74%

Section: Introduction and Resultsmentioning

confidence: 96%

On a refinement of the generalized Catalan numbers for Weyl groups

Athanasiadis

2004

Trans. Amer. Math. Soc.

127

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“…On the other hand, Cellini and Papi [38] gave a bijective proof that the positive chambers in the Catalan arrangement are counted by (1.1). They defined two bijections: one from the W -orbits in Q/(h + 1)Q to the ad-nilpotent ideals in a Borel subalgebra of the corresponding semisimple Lie algebra, and another from these ideals to antichains in the root poset.…”

Section: Symmetries Of Regular Polytopesmentioning

confidence: 99%

“…Next, we discuss the Shi hyperplane arrangement and its relation to nonnesting partitions by Cellini and Papi [38]. Athanasiadis [3,4] has defined a class…”

Section: 22) We Note That Bessis and Corran Have Independently Comentioning

confidence: 99%

Generalized noncrossing partitions and combinatorics of Coxeter groups

2009

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under whose guidance the original draft [2] was written. I thank my wife Heather for her thorough reading of the manuscript and for helpful mathematical discussions. Thanks to the anonymous referee for many valuable suggestions. And thanks to Hugh Thomas who made several contributions to my understanding; in particular, he suggested the proof of Theorem 3.7.2.I have benefitted from valuable conversations with many mathematicians; their suggestions have improved this memoir in countless ways. Thanks to: (in alphabetical order)

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“…Since then, abelian ideals attracted a lot of attention, see e.g. [2,3,4,8,9,12,15]. We think of Ab as a poset with respect to inclusion.…”

Section: Introductionmentioning

confidence: 99%

Abelian ideals of a Borel subalgebra and root systems

Panyushev¹

2014

J. Eur. Math. Soc.

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ABSTRACT. Let g be a simple Lie algebra and Ab o the poset of non-trivial abelian ideals of a fixed Borel subalgebra of g. In [9], we constructed a partition Ab o = ⊔ µ Ab µ parameterised by the long positive roots of g and studied the subposets Ab µ . In this note, we show that this partition is compatible with intersections, relate it to the Kostant-Peterson parameterisation and to the centralisers of abelian ideals. We also prove that the poset of positive roots of g is a join-semilattice.

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Ad-nilpotent ideals of a Borel subalgebra II

Cited by 75 publications

References 14 publications

On a refinement of the generalized Catalan numbers for Weyl groups

On a refinement of the generalized Catalan numbers for Weyl groups

Generalized noncrossing partitions and combinatorics of Coxeter groups

Abelian ideals of a Borel subalgebra and root systems

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