# Lorentzian Lattices and E-Polytopes

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Root Lattices and Hyperplanes

n | 3 | 4 | 5 | 6 | 7 | 8 |

$|{\mathbf{R}}_{n}|$ | 12 | 20 | 40 | 72 | 126 | 240 |

Total numbers of the roots of ${\mathbf{R}}_{n}$ |

n | 3 | 4 | 5 | 6 | 7 | 8 |

E_{n} | A_{1} × A_{2} | A_{4} | D_{5} | E_{6} | E_{7} | E_{8} |

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Definition**

**1.**

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Corollary**

**1.**

## 3. Fundamental Lattice Vectors

#### 3.1. Lines

n | 3 | 4 | 5 | 6 | 7 | 8 |

$|{\mathbf{L}}_{n}|$ | 6 | 10 | 16 | 27 | 56 | 240 |

Total numbers of the lines |

**Theorem**

**2**

**.**The lines of ${\mathbf{L}}_{n}$ correspond bijectively to the vertices of a Gosset polytope ${(n-4)}_{21}$ in ${\Lambda}_{n}^{1}\otimes \mathbb{Q}$.

**Remark**

**2.**

#### 3.2. Rational Conic Vectors

n | 3 | 4 | 5 | 6 | 7 | 8 |

$|{\mathbf{Con}}_{n}|$ | 3 | 5 | 10 | 27 | 126 | 2160 |

Total numbers of the rational conics |

**Lemma**

**3**

**.**Let a and l be a rational conic vector and a line in ${\mathbb{Z}}^{n+1}$, respectively. Then, one has: (1) The line l corresponds to a vertex of the $(n-1)$-crosspolytope associated with a if and only if $(l,a)=0$. (2) Assume $a={l}_{1}^{a}+{l}_{2}^{a}$ where ${l}_{1}^{a}$ and ${l}_{2}^{a}$ are lines. Then the line $l$ corresponds to a vertex of the $(n-1)$-crosspolytope associated with a if and only if $(l,{l}_{1}^{a})=(l,{l}_{2}^{a})=0$.

**Remark**

**3.**

#### 3.3. Rational Cubic Vectors

n | 3 | 4 | 5 | 6 | 7 | 8 |

$|{\mathbf{Cub}}_{n}|$ | 2 | 5 | 15 | 72 | 576 | 17520 |

Total numbers of the rational cubics |

**Lemma**

**4**

**.**Let $b$ be a rational cubic vector in ${\mathbb{Z}}^{n+1}$. One has: (1) If $3\le n\le 7$, then $3b+{K}_{n}$ can be written as a sum of $n$ lines ${l}_{1},{l}_{2},\cdots {l}_{n}$ with $({l}_{i},{l}_{j})=0$ for $i\ne j$. Conversely, for each configuration ${l}_{1},{l}_{2},\cdots {l}_{n}$ of mutually orthogonal lines,

## 4. Line Configurations

#### 4.1. Lattice Pairings of Lines

n | 3 | 4 | 5 | 6 | 7 | 8 |

$|{A}_{n}|$ | 9 | 30 | 80 | 216 | 756 | 6720 |

Total numbers of the edges in Gosset polytopes (n − 4)_{21} |

#### 4.2. Line Hierarchy and the Gosset Polytope ${(n-4)}_{21}$

n | 3 | 4 | 5 | 6 | 7 | 8 |

$|{B}_{0}^{n}\left(l\right)|$ | 3 | 6 | 10 | 16 | 27 | 56 |

n | 3 | 4 | 5 | 6 | 7 | 8 |

$|{B}_{1}^{n}\left(l\right)|$ | 2 | 3 | 5 | 10 | 27 | 126 |

#### 4.3. Roots as Configurations of Orthogonal Lines

**Theorem**

**3.**

**Proof.**

**Remark**

**4.**

**Corollary**

**2.**

**Proof.**

**Remark**

**5.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Clingher, A.; Lee, J.-H.
Lorentzian Lattices and E-Polytopes. *Symmetry* **2018**, *10*, 443.
https://doi.org/10.3390/sym10100443

**AMA Style**

Clingher A, Lee J-H.
Lorentzian Lattices and E-Polytopes. *Symmetry*. 2018; 10(10):443.
https://doi.org/10.3390/sym10100443

**Chicago/Turabian Style**

Clingher, Adrian, and Jae-Hyouk Lee.
2018. "Lorentzian Lattices and E-Polytopes" *Symmetry* 10, no. 10: 443.
https://doi.org/10.3390/sym10100443