Root Polytopes and Borel Subalgebras

Cellini, Paola; Marietti, Mario

doi:10.1093/imrn/rnu070

Cited by 12 publications

(66 citation statements)

References 10 publications

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“…Note that the result is precisely [CM,Theorem 4.5 (1) ⇔ (2)]. We now show that it quickly follows from the analysis in previous sections, via Proposition 8.2.…”

Section: Example: Finite-dimensional Representations Over a Simple LIsupporting

confidence: 69%

“…Apart from the length, the longest weights generalize to V λ several properties satisfied by the long roots in L(θ) = g for simple g. For instance, Propositions 7.1 and 7.2 extend [CM,Proposition 3.3(1), Remark 3.8, Lemma 3.12, and Propositions 3.9 and 3.16] to all highest weight modules V λ .…”

Section: Half-space Representation and Facetsmentioning

confidence: 95%

“…For instance, the size of the set V J = wt I\J L(θ) is a computation specific to the root system and is expressed in terms of other root systems; see [CM,Theorem 1.1(1)]. However, there are other statements that are specific to the adjoint representation and yet can be obtained from our results in previous sections via Proposition 8.2.…”

Section: Example: Finite-dimensional Representations Over a Simple LImentioning

confidence: 96%

“…Now given J ⊂ I, [CM,Proposition 3.7] says that F J ′ = F J if and only if ∂J ⊂ J ′ ⊂ J. On the other hand, Theorem A implies in the special case V λ = M (θ, I) = g that wt J ′ g = wt J g ⇐⇒ J min ⊂ J ′ ⊂ J max .…”

Section: Example: Finite-dimensional Representations Over a Simple LImentioning

confidence: 99%

“…The second assertion follows from equation (8.3). To prove the first assertion, we first recall the definition of ∂J and J from [CM,Section 3]. Let I := I ⊔{0} correspond to the simple roots ∆ := ∆ ⊔ {α 0 } associated to the affine Lie algebra g. Now given J ⊂ I, let ( I \ J) 0 denote the connected component of the Dynkin diagram of I \ J that contains the affine root α 0 , and define: [Bou,Chapter VI.4.3] that in the Dynkin diagram for g, the affine root α 0 can be viewed as the negative of the highest root for simple g: α 0 = −θ.…”

Section: Example: Finite-dimensional Representations Over a Simple LImentioning

confidence: 99%

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Standard parabolic subsets of highest weight modules

Khare

2016

Trans. Amer. Math. Soc.

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Abstract. In this paper we study certain fundamental and distinguished subsets of weights of an arbitrary highest weight module over a complex semisimple Lie algebra. These sets wt J V λ are defined for each highest weight module V λ and each subset J of simple roots; we term them "standard parabolic subsets of weights". It is shown that for any highest weight module, the sets of simple roots whose corresponding standard parabolic subsets of weights are equal form intervals in the poset of subsets of the set of simple roots under containment. Moreover, we provide closed-form expressions for the maximum and minimum elements of the aforementioned intervals for all highest weight modules V λ over semisimple Lie algebras g. Surprisingly, these formulas only require the Dynkin diagram of g and the integrability data of V λ . As a consequence, we extend classical work by Satake, Borel-Tits, Vinberg, and Casselman, as well as recent variants by Cellini-Marietti to all highest weight modules.We further compute the dimension, stabilizer, and vertex set of standard parabolic faces of highest weight modules, and show that they are completely determined by the aforementioned closed-form expressions. We also compute the f -polynomial and a minimal half-space representation of the convex hull of the set of weights. These results were recently shown for the adjoint representation of a simple Lie algebra, but analogues remain unknown for any other finite-or infinite-dimensional highest weight module. Our analysis is uniform and type-free, across all semisimple Lie algebras and for arbitrary highest weight modules.

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“…Note that the result is precisely [CM,Theorem 4.5 (1) ⇔ (2)]. We now show that it quickly follows from the analysis in previous sections, via Proposition 8.2.…”

Section: Example: Finite-dimensional Representations Over a Simple LIsupporting

confidence: 69%

Section: Half-space Representation and Facetsmentioning

confidence: 95%

Section: Example: Finite-dimensional Representations Over a Simple LImentioning

confidence: 96%

Section: Example: Finite-dimensional Representations Over a Simple LImentioning

confidence: 99%

Section: Example: Finite-dimensional Representations Over a Simple LImentioning

confidence: 99%

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Standard parabolic subsets of highest weight modules

Khare

2016

Trans. Amer. Math. Soc.

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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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Root polytope and partitions

Chirivì

2014

J Algebr Comb

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Given a crystallographic reduced root system and an element γ of the lattice generated by the roots, we study the minimum number |γ|, called the length of γ, of roots needed to express γ as sum of roots. This number is related to the linear functionals presenting the convex hull of the roots. The map γ −→ |γ| turns out to be the upper integral part of a piecewise linear function with linearity domains the cones over the facets of this convex hull. In order to show this relation we investigate the integral closure of the monoid generated by the roots in a facet. We study also the positive length, i.e. the minimum number of positive roots needed to write an element, and we prove that the two notions of length coincide only for the types A ℓ and C ℓ .

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Root Polytopes and Borel Subalgebras

Cited by 12 publications

References 10 publications

Standard parabolic subsets of highest weight modules

Standard parabolic subsets of highest weight modules

Root polytopes and Abelian ideals

Root polytope and partitions

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