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

Publisher Copyright:
© 2018 by the authors.

Keywords

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

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10.3390/sym10100443

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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.",

keywords = "Dual lattice, E-polytope, Gosset polytope, Lines, Lorentzian lattice, Root lattice, Weyl group",

author = "Adrian Clingher and Lee, \{Jae Hyouk\}",

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year = "2018",

doi = "10.3390/sym10100443",

language = "English",

volume = "10",

journal = "Symmetry",

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AU - Clingher, Adrian

AU - Lee, Jae Hyouk

PY - 2018

Y1 - 2018

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

UR - https://www.scopus.com/pages/publications/85055752205

U2 - 10.3390/sym10100443

DO - 10.3390/sym10100443

M3 - Article

AN - SCOPUS:85055752205

SN - 2073-8994

VL - 10

JO - Symmetry

JF - Symmetry

IS - 10

M1 - 443

ER -

Lorentzian lattices and E-polytopes

Abstract

Bibliographical note

Keywords

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