Lattices from group frames and vertex transitive graphs (Lenny Fukshansky, CMC)

Tight frames in Euclidean spaces are widely used convenient generalizations of orthonormal bases. A particularly nice class of such frames is generated as orbits under irreducible actions of finite groups of orthogonal matrices: these are called irreducible group frames. Integer spans of rational irreducible group frames form Euclidean lattices with some very nice geometric properties, called strongly eutactic lattices. We discuss this construction, focusing on an especially interesting infinite family in arbitrarily large dimensions, which comes from vertex transitive graphs. We demonstrate several examples of such lattices from graphs that exhibit some rather fascinating properties. This is joint work with D. Needell, J. Park and J. Xin.