A pentagonal crystal, the golden section, alcove packing and aperiodic tilings

Joseph, Anthony

doi:10.1007/s00031-009-9064-y

Cited by 4 publications

(1 citation statement)

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“…Similarly to Remark 1.3, our constructions, results and conjectures make sense if one replaces U q (g) by U(g) for any (symmetrizable or not) Kac-Moody algebra g. Some results (for example, Theorem 1.8) should be possible to rescue even when W is not crystallographic (and so g does not exists) with the aid of theory of continuous crystals initiated by A. Joseph in [22].…”

Section: Introductionmentioning

confidence: 73%

On Cacti and Crystals

Berenstein

Greenstein

Lǐ

2019

Representations and Nilpotent Orbits of Lie Algebraic Systems

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In the present work we study actions of various groups generated by involutions on the category O int q (g) of integrable highest weight U q (g)-modules and their crystal bases for any symmetrizable Kac-Moody algebra g. The most notable of them are the cactus group and (yet conjectural) Weyl group action on any highest weight integrable module and its lower and upper crystal bases. Surprisingly, some generators of cactus groups are anti-involutions of the Gelfand-Kirillov model for O int q (g) closely related to the remarkable quantum twists discovered by Kimura and Oya in [28].The following Lemmata are apparently well-known. We provide their proof for the reader's convenience.Proof. The argument is by induction on d. The case d = 0 (that is,By Lemma 2.5, K J = {w ∈ W : uwu −1 ∈ W J , ∀ u ∈ W } and is a subgroup of W J . Suppose that w ∈ K J ; in particular, w ∈ W J . Furthermore, using Lemma 2.7 with u = w and i ∈ I \ J, we conclude that w ∈ i∈I\J W {i} ⊥ = W (I\J) ⊥ . Let i ′ ∈ ∂(I \ J). By definition, there exists i ∈ I \ J and an admissible sequence (i 0 , . . . , i d ) with d = dist(i, i ′ ), i 0 = i and i d = i ′ . Since uwu −1 ∈ W J with u = s i 0 · · · s i d , it follows from Lemma 2.8 4.3. Parabolic involutions and proof of Theorem 1.5. In view of Theorem 4.10,for any λ J ∈ P + J , β ∈ P and v ∈ I J λ J (V )(β). Note that I J λ J (V )(β) = 0 unless λ J − β ∈ j∈J Z ≥0 α j . We need the following properties of σ J .Proposition 4.16. Let J, J ′ ∈ J be orthogonal. Then σ J⊔J ′ = σ J • σ J ′ .

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Section: Introductionmentioning

confidence: 73%