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].

The geometry of H4 polytopes

We describe the geometry of an arrangement of 24-cells inscribed in the 600-cell. In Section 7 we apply our results to the even unimodular lattice E8 and show how the 600-cell transforms E8/2E8, an 8-space over the field F2, into a 4-space over F4 whose points, lines and planes are labeled by the geometric objects of the 600-cell.

The theta functions of sublattices of the Leech lattice

Let Λ be the Leech lattice which is an even unimodular lattice with no vectors of squared length 2 in 24-dimensional Euclidean space R24. Then the Mathieu Group M24 is a subgroup of the automorphism group .0 of Λ and the action on Λ of M24 induces a natural permutation representation of M24 on an orthogonal basis For , let Λm be the sublattice of vectors invariant under m:

Leech roots and Vinberg groups

We study the recently defined Leech roots, which have many remarkable properties. They are the fundamental roots for the even unimodular lattice in Lorentzian space R 25,1 , and correspond one for one with the points of the Leech lattice. The paper contains an extensive table of the Leech roots in both Euclidean and hyperbolic coordinates. We provide the first of what promise to be many applications by showing that the Leech roots simplify and explain the remarkable results of Vinberg and Kaplinskaja on the reflexion groups of unimodular Lorentzian lattices in dimensions below 20. They also enable us to make some progress on the study of these groups in the next few dimensions.