Root product of lattices

Root Lattices and Quasicrystals

It is shown how root lattices and their reciprocals might serve as the right pool for the construction of quasicrystalline structure models. All non-periodic symmetries observed so far are covered in minimal embedding with maximal symmetry. For the construction of quasiperiodic tiling models by means of projection from higherdimensional periodic structures, the primitive hypercubic lattices are...

A characterization of root lattices

Cyclotomic structures on root lattices

The centralizer C(w) of an element w in a Weyl group W plays an important role in the structure and representation theory of split reductive groups G over finite and p-adic fields k, where W is the absolute Weyl group of G. If k is finite, this is well-known: the element w determines a maximal ktorus Tw ⊂ G and C(w) may be identified with the k-rational points in the Weyl group W (Tw, G) of Tw ...

Quotient Lattices Modulo Filters and Direct Product of Two Lattices

Binary and unary operation preserving binary relations and quotients of those operations modulo equivalence relations are introduced. It is shown that the quotients inherit some important properties (commutativity, associativity, distributivity, ect.). Based on it the quotient (also called factor) lattice modulo filter (ie. modulo the equivalence relation w.r.t the filter) is introduced. Simila...

Root Polytopes and Growth Series of Root Lattices

The convex hull of the roots of a classical root lattice is called a root polytope. We determine explicit unimodular triangulations of the boundaries of the root polytopes associated to the root lattices An, Cn and Dn, and compute their f -and h-vectors. This leads us to recover formulae for the growth series of these root lattices, which were first conjectured by Conway–Mallows–Sloane and Baak...