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Root lattice

Description

A lattice that is symmetric under a specific crystallographic reflection group; see [1; Table 4.1] for the list of crystallographic reflection groups and their corresponding root lattices. The root-lattice family consists of lattices \(A_n\), \(\mathbb{Z}^n\), or \(D_n\) for dimension \(n\), or \(E_{i}\) for \(i\in\{6,7,8\}\). Their generator matrices can be taken to be the root matrices of the corresponding reflection groups.

Protection

The densest lattice packings in dimensions \(3\) through \(8\) are root lattices [1; Table 1.1].

Member of code lists

Primary Hierarchy

Classical Domain
Analog Kingdom
Analog codeECC
Sphere packing
LatticeECC
Parents
Lattice
Root lattice
Children
\(E_8\) Gosset lattice
\(\mathbb{Z}^n\) hypercubic lattice
\(A_n\) lattice
\(D_n\) checkerboard lattice
\(E_7\) root lattice
\(E_6\) root lattice

References

[1]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
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Zoo Code ID: root

Cite as:
“Root lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/root, arXiv:2606.11484
BibTeX:
@incollection{eczoo_root,
title={Root lattice},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/root}
}
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Permanent link:
https://errorcorrectionzoo.org/c/root

Cite as:

“Root lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/root, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/analog/sphere_packing/lattice/root/root.yml.