Lorentzian lattices and E-polytopes
We consider certain E -type root lattices embedded within the standard Lorentzian lattice ℤ (3 ≤ n ≤ 8) and study their discrete geometry from the point of view of del Pezzo surface geometry. The lattice ℤ 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) . n 21 n+1 n+1
Clingher, Adrian and Lee, Jae Hyouk, "Lorentzian lattices and E-polytopes" (2018). Mathematics and Statistics Faculty Works. 1.
Available at: https://irl.umsl.edu/mathstats-faculty/1