Lorentzian lattices and E-polytopes

Adrian Clingher, Jae Hyouk Lee

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Article number443
Issue number10
StatePublished - 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.


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


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