Valuative invariants for polymatroids

Many important invariants for matroids and polymatroids, such as the Tutte polynomial, the Billera-Jia-Reiner quasi-symmetric function, and the invariant $\mathcal G$ introduced by the first author, are valuative. In this paper we construct the $\Z$-modules of all $\Z$-valued valuative functions for labeled matroids and polymatroids on a fixed ground set, and their unlabeled counterparts, the $\Z$-modules of valuative invariants. We give explicit bases for these modules and for their dual modules generated by indicator functions of polytopes, and explicit formulas for their ranks. Our results confirm a conjecture of the first author that $\mathcal G$ is universal for valuative invariants.Comment: 54 pp, 9 figs. Mostly minor changes; Cor 10.5 and formula for products of $u$s corrected; Prop 7.2 is new. To appear in Advances in Mathematic

About the kernel of the augmentation of finitely generated Z-modules

Let M be a free finitely generated Z-module with basis B and ΔM the kernel of the homomorphism M→Z which maps B to 1. A basis of ΔM can be easily constructed from the basis B of M. Let further R be a submodule of M such that N = M/R is free. The subject of investigation is the module ΔN = (ΔM + R) / R. We compute the index [N:ΔN] and construct bases of ΔN with the help of a basis of N. Finally, the results are applied to a special class of modules which is connected with the group of cyclotomic units

Coincidence site modules in 3-space

The coincidence site lattice (CSL) problem and its generalization to Z-modules in Euclidean 3-space is revisited, and various results and conjectures are proved in a unified way, by using maximal orders in quaternion algebras of class number 1 over real algebraic number fields.Comment: 25 page

Invariant Submodules and Semigroups of Self-Similarities for Fibonacci Modules

The problem of invariance and self-similarity in Z-modules is investigated. For a selection of examples relevant to quasicrystals, especially Fibonacci modules, we determine the semigroup of self-similarities and encapsulate the number of similarity submodules in terms of Dirichlet series generating functions.Comment: 7 pages; to appear in: Aperiodic 97, eds. M. de Boissieu, J. L. Verger-Gaugry and R. Currat, World Scientific, Singapore (1998), in pres