Lorentzian lattices and E-polytopes

Adrian Clingher; Jae Hyouk Lee

doi:10.3390/sym10100443

Lorentzian lattices and E-polytopes

Adrian Clingher, Jae Hyouk Lee

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We consider certain E_n-type root lattices embedded within the standard Lorentzian lattice ℤⁿ⁺¹ (3 ≤ n ≤ 8) and study their discrete geometry from the point of view of del Pezzo surface geometry. The lattice ℤⁿ⁺¹ decomposes as a disjoint union of affine hyperplanes which satisfy a certain periodicity. We introduce the notions of line vectors, rational conic vectors, and rational cubics vectors and their relations to E-polytopes. We also discuss the relation between these special vectors and the combinatorics of the Gosset polytopes of type (n - 4)₂₁.

Original language	English
Article number	443
Journal	Symmetry
Volume	10
Issue number	10
DOIs	https://doi.org/10.3390/sym10100443
State	Published - 2018

Bibliographical note

Funding Information:
Funding: The first author was supported by a University of Missouri Research Board Grant and the second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No.2016R1D1A1B03931673).

Publisher Copyright:
© 2018 by the authors.

Keywords

Dual lattice
E-polytope
Gosset polytope
Lines
Lorentzian lattice
Root lattice
Weyl group

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title = "Lorentzian lattices and E-polytopes",

abstract = "We consider certain En-type root lattices embedded within the standard Lorentzian lattice ℤn+1 (3 ≤ n ≤ 8) and study their discrete geometry from the point of view of del Pezzo surface geometry. The lattice ℤn+1 decomposes as a disjoint union of affine hyperplanes which satisfy a certain periodicity. We introduce the notions of line vectors, rational conic vectors, and rational cubics vectors and their relations to E-polytopes. We also discuss the relation between these special vectors and the combinatorics of the Gosset polytopes of type (n - 4)21.",

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N2 - We consider certain En-type root lattices embedded within the standard Lorentzian lattice ℤn+1 (3 ≤ n ≤ 8) and study their discrete geometry from the point of view of del Pezzo surface geometry. The lattice ℤn+1 decomposes as a disjoint union of affine hyperplanes which satisfy a certain periodicity. We introduce the notions of line vectors, rational conic vectors, and rational cubics vectors and their relations to E-polytopes. We also discuss the relation between these special vectors and the combinatorics of the Gosset polytopes of type (n - 4)21.

AB - We consider certain En-type root lattices embedded within the standard Lorentzian lattice ℤn+1 (3 ≤ n ≤ 8) and study their discrete geometry from the point of view of del Pezzo surface geometry. The lattice ℤn+1 decomposes as a disjoint union of affine hyperplanes which satisfy a certain periodicity. We introduce the notions of line vectors, rational conic vectors, and rational cubics vectors and their relations to E-polytopes. We also discuss the relation between these special vectors and the combinatorics of the Gosset polytopes of type (n - 4)21.

KW - Dual lattice

KW - E-polytope

KW - Gosset polytope

KW - Lines

KW - Lorentzian lattice

KW - Root lattice

KW - Weyl group

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JO - Symmetry

JF - Symmetry

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Lorentzian lattices and E-polytopes

Abstract

Bibliographical note

Keywords

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