Root polytope and partitions

Chirivì, Rocco

doi:10.1007/s10801-014-0526-5

Cited by 5 publications

(4 citation statements)

References 14 publications

(36 reference statements)

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“…This implies in particular that, for Φ of type A n or C n , P + is the intersection of P with the positive cone generated by the positive roots, which is false in general. Our result has a direct application in the recent paper [10] on partition functions.…”

Section: Introductionmentioning

confidence: 63%

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Root polytopes and Abelian ideals

Cellini

Marietti

2013

J Algebr Comb

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We study the root polytope P Φ of a finite irreducible crystallographic root system Φ using its relation with the abelian ideals of a Borel subalgebra of a simple Lie algebra with root system Φ. We determine the hyperplane arrangement corresponding to the faces of codimension 2 of P Φ and analyze its relation with the facets of P Φ . For Φ of type A n or C n , we show that the orbits of some special subsets of abelian ideals under the action of the Weyl group parametrize a triangulation of P Φ . We show that this triangulation restricts to a triangulation of the positive root polytope P + Φ .

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Section: Introductionmentioning

confidence: 63%

“…Theorem 5.8 has a direct application in [10]. As noted in [10], P + = P ∩ C + for all root types other than A and C.…”

Section: Principal Maximal Abelian Ideals Of the Borel Subalgebramentioning

confidence: 96%

Root polytopes and Abelian ideals

Cellini

Marietti

2013

J Algebr Comb

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“…This is one of the special properties of the root politope that hold only for the types A and C (see also [7]). In fact, it is easy to see that, for all other root types, P `is properly contained in P X ConepΦ `q [4]. Hence, in these cases, from the standard parabolic facets, we cannot obtain any triangulation of the positive root polytope.…”

Section: Discussionmentioning

confidence: 99%

Triangulations of root polytopes

Cellini¹

2016

Preprint

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“…This is one of the special properties of the root polytope that hold only for the types A and C (see also [5]). Indeed, it is easy to see that, for all other root types, P + is properly contained in P ∩Cone(Φ + ) [10]. Hence, in these cases, from a triangulation of the standard parabolic facets, we cannot obtain a triangulation of the positive root polytope in a natural way.…”

Section: Discussionmentioning

confidence: 99%

Triangulations of root polytopes

Cellini¹

2018

Algebraic Combinatorics

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Let Φ be an irreducible crystallographic root system and P its root polytope, i.e., the convex hull of Φ. We provide a uniform construction, for all root types, of a triangulation of the facets of P. We also prove that, on each orbit of facets under the action of the Weyl group, the triangulation is unimodular with respect to a root sublattice that depends on the orbit.Definition 5.1. Let J ⊆ I. We say that J is bipartite if it has an initial section J i , and a final section J f such thatIf the above conditions hold, we say that (3) is called a separating hyperplane, for the bipartitionNote that, by definition, if J has a proper bipartition, then it has at least two elements. If J is also saturated, then it contains two crossing pairs and these provide at least three elements in J. The definition also implies that, if {J i , J f } is a bipartition of J, then J i J f and J f J i are an initial and a final section of J, respectively, since the complement of an initial section is a final section, and vice-versa, and we havewhere by we denote disjoint union. Finally, we note that if J is saturated, also all the subsetsDefinition 5.2. For each subset S of Φ + , we define the restricted relations S and ∼ S on S as follows. For all β 1 , β 2 ∈ S we set: (1) β 1 S β 2 if there exists a middle pair {γ 1 , γ 2 } between β 1 and β 2 contained in S;Obviously, for any S ⊆ Φ + and β 1 , β 2 ∈ S, the relation β 1 S β 2 implies β 1 β 2 . Hence, if S is ∼closed, then, for all β 1 , β 2 ∈ S, we have β 1 ∼ β 2 if and only if β 1 ∼ S β 2 .The first of following lemmas is clear, hence we omit the proof.Lemma 5.3. Let S ⊆ Φ + . If S is saturated, then S is ∼closed.Lemma 5.4. Let I be an abelian ideal in Φ, Ψ a root subsystem of Φ, and Ψ 1 , . . . , Ψ k be the irreducible components of Ψ. Moreover, let J be a ∼closed subset of Φ such that J ⊆ I ∩ Ψ, and let R ⊆ J. Then, R is reduced in Φ if and only if R ∩ Ψ i is reduced in Ψ i for all i ∈ {i, . . . , k}.

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Root polytope and partitions

Cited by 5 publications

References 14 publications

Root polytopes and Abelian ideals

Root polytopes and Abelian ideals

Triangulations of root polytopes

Triangulations of root polytopes

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