Root lattices in number fields

We explore whether a root lattice may be similar to the lattice [Formula: see text] of integers of a number field [Formula: see text] endowed with the inner product [Formula: see text], where [Formula: see text] is an involution of [Formula: see text]. We classify all pairs [Formula: see text], [Formula: see text] such that [Formula: see text] is similar to either an even root lattice or the root lattice [Formula: see text]. We also classify all pairs [Formula: see text], [Formula: see text] such that [Formula: see text] is a root lattice. In addition to this, we show that [Formula: see text] is never similar to a positive-definite even unimodular lattice of rank [Formula: see text], in particular, [Formula: see text] is not similar to the Leech lattice. In Appendix B, we give a general cyclicity criterion for the primary components of the discriminant group of [Formula: see text].

Generalized Dual-Root Lattice Transforms of Affine Weyl Groups

Discrete transforms of Weyl orbit functions on finite fragments of shifted dual root lattices are established. The congruence classes of the dual weight lattices intersected with the fundamental domains of the affine Weyl groups constitute the point sets of the transforms. The shifted weight lattices intersected with the fundamental domains of the extended dual affine Weyl groups form the sets of labels of Weyl orbit functions. The coinciding cardinality of the point and label sets and corresponding discrete orthogonality relations of Weyl orbit functions are demonstrated. The explicit counting formulas for the numbers of elements contained in the point and label sets are calculated. The forward and backward discrete Fourier-Weyl transforms, together with the associated interpolation and Plancherel formulas, are presented. The unitary transform matrices of the discrete transforms are exemplified for the case A 2 .

Lattices From Tight Frames and Vertex Transitive Graphs

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.

Lorentzian Lattices and E-Polytopes

We consider certain E n -type root lattices embedded within the standard Lorentzian lattice Z n + 1 ( 3 ≤ n ≤ 8 ) and study their discrete geometry from the point of view of del Pezzo surface geometry. The lattice Z n + 1 decomposes as a disjoint union of affine hyperplanes which satisfy a certain periodicity. We introduce the notions of line vectors, rational conic vectors, and rational cubics vectors and their relations to E-polytopes. We also discuss the relation between these special vectors and the combinatorics of the Gosset polytopes of type ( n − 4 ) 21 .