Claremont McKenna College, Mathematics (CMC), Pomona College, Mathematics (Pomona)
We consider the set of all linear combinations with integer coefficients of the vectors of a unit tight equiangular (k,n) frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the k-dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a lattice for n=k+1 and that there are infinitely many k such that a lattice emerges for n=2k. We dispose of all cases in dimensions k at most 9. In particular, we show that a (7,28) frame generates a strongly eutactic lattice and give an alternative proof of Roland Bacher's recent observation that this lattice is perfect.
© 2016 Elsevier Inc.
Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj and Deanna Needell. “Lattices from equiangular tight frames.” Linear Algebra and its Applications, vol. 510, 395–420, 2016.