Codes and Expansions (CodEx) Seminar

Frank Vallentin (Universität zu Köln)
Coloring the Voronoi tessellation of lattices

In this talk I will define the chromatic number of a lattice: It is the least number of colors one needs to color the interiors of the cells of the Voronoi tessellation of a lattice so that no two cells sharing a facet are of the same color.
Then I will consider the chromatic number of several important lattices: the root lattices and the Leech lattice. For this I will apply a spectral lower bound for the chromatic number of lattices in spirit of Hoffman's bound for finite graphs. This bound will be computed for the root lattices by relating it to the character theory of the corresponding Lie groups.
Using the relation to sphere packing I will investigate the asymptotic behaviour of the chromatic number of lattices when the dimension tends to infinity.
(based on joint work with David Madore, Mathieu Dutour Sikirić, and Philippe Moustrou, arXiv:1907.09751 [math.CO])