Abstract
We show that real tight frames that generate lattices must be rational. In the case of irreducible group frames, we show that the corresponding lattice is always strongly eutactic. We use this observation to describe a construction of strongly eutactic lattices from vertex transitive graphs. We show that such lattices exist in arbitrarily large dimensions and discuss some examples arising from complete graphs, product graphs, as well as some other notable examples of graphs. In particular, some well-known root lattices and those related to them can be recovered this way. We discuss various properties of this construction and also mention some potential applications of lattices generated by incoherent systems of vectors.
Highlights
Let, be the usual inner product on Rk and x := x, x 1/2 the Euclidean norm on Rk
A interesting class of lattices are eutactic lattices: a lattice L is called eutactic if its set of minimal vectors S(L) satisfies a eutaxy condition, i.e. there exist positive real numbers c1, . . . , cn, such that v 2=
If some of the ci’s are irrational, Q is a form with irrational coefficients. In view of this observation, it is interesting to understand what are the necessary and sufficient conditions on a k × n real matrix B so that BZn is a lattice to imply that B must be rational? In the rest of this section we prove a sufficient condition that is weaker than being a tight frame
Summary
Let , be the usual inner product on Rk and x := x, x 1/2 the Euclidean norm on Rk. For a lattice L ⊂ Rk of full rank k (that is a discrete co-compact subgroup of Rk) the minimal norm of L is. We can say that a lattice is strongly eutactic whenever its set of minimal vectors forms a uniform tight frame. Let G be a group of k × k real orthogonal matrices and f ∈ Rk be a vector so that F = Gf is an irreducible rational group frame in Rk. the lattice L(F) is strongly eutactic. Implies more, that in our setting (in the case of distance transitive graphs) these vectors generate a lattice whose set of minimal vectors is strongly eutactic.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
