Almost impossible \(E_8\) and Leech lattices
Summary: We start this short note by introducing two remarkable mathematical objects: the \(E_8\) root lattice \(\Lambda_8\) in 8-dimensional Euclidean space and the Leech lattice \(\Lambda_{24}\) in 24-dimensional space. These two lattices stand out among their lattice sisters for several reasons. The first reason is that these both lattices are related to other unique and exceptional mathematical objects. The \(E_8\) lattice is the root lattice of the semisimple exceptional Lie algebra \(E_8\). The quotient of \(\Lambda_8\) by a suitable sublattice is isomorphic to the Hamming binary code of dimension 8 and minimum distance 4, which in its turn is an optimal error-correcting binary code with these parameters. The Leech lattice is famously connected to the exceptional finite simple groups, monstrous moonshine [\textit{J. H. Conway} and \textit{S. P. Norton}, Bull. Lond. Math. Soc. 11, 308--339 (1979; Zbl 0424.20010)] and the monster vertex algebra [\textit{R. E. Borcherds}, Invent. Math. 109, No. 2, 405--444 (1992; Zbl 0799.17014)]. Another reason is that \(\Lambda_8\) and \(\Lambda_{24}\) are solutions to a number of optimization problems. The \(E_8\) and Leech lattice provide optimal sphere packings in their respective dimensions [\textit{H. Cohn} et al., Ann. Math. (2) 185, No. 3, 1017--1033 (2017; Zbl 1370.52037); \textit{M. S. Viazovska}, ibid. 185, No. 3, 991--1015 (2017; Zbl 1373.52025)]. Also both lattices are universally optimal, which means that among all point configurations of the same density, the \(\Lambda_8\) and \(\Lambda_{24}\) have the smallest possible Gaussian energy [\textit{H. Cohn} et al., ``Universal optimality of the \(E_8\) and Leech lattices and interpolation formulas'', Preprint, \url{arXiv:1902.05438}]. The third reason for our interest in these lattices is less obvious. The optimality of the \(E_8\) and Leech lattices can be proven in a rather short way, while the solutions of analogous problems in other dimensions, even dimensions much smaller than 8 and 24, is still wide open. Finally, this last property seems to be inherited by other geometric objects obtained from \(\Lambda_{8}\) and \(\Lambda_{24}\), such as Hamming code, Golay code and the sets of shortest vectors of both lattices.
scientific article; zbMATH DE number 7475672
scientific article