Sphere Packings, Lattices and Groups

Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

Groupes et algébres de lie

This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many clusters. This classification turns out to be identical to the Cartan-Killing classification of semisimple Lie algebras and finite root systems, which is intriguing since in most cases, the symmetry exhibited by the Cartan-Killing type of a cluster algebra is not at all apparent from its geometric origin. 
The combinatorial structure behind a cluster algebra of finite type is captured by its cluster complex. We identify this complex as the normal fan of a generalized associahedron introduced and studied in hep-th/0111053 and math.CO/0202004. Another essential combinatorial ingredient of our arguments is a new characterization of the Dynkin diagrams.

Cluster algebras II: Finite type classification

regular polytopes stand at the end of more than two millennia of geometrical research, which began with regular polygons and polyhedra. They are highly symmetric combinatorial structures with distinctive geometric, algebraic or topological properties; in many ways more fascinating than traditional regular polytopes and tessellations. The rapid development of the subject in the past 20 years has resulted in a rich new theory, featuring an attractive interplay of mathematical areas, including geometry, combinatorics, group theory and topology. regular polytopes and their groups provide an appealing new approach to understanding geometric and combinatorial symmetry. This is the first comprehensive up-to-date account of the subject and its ramifications, and meets a critical need for such a text, because no book has been published in this area of classical and modern discrete geometry since Coxeter's Regular Polytopes (1948) and Regular Complex Polytopes (1974). The book should be of interest to researchers and graduate students in discrete geometry, combinatorics and group theory.

https://assets.cambridge.org/052181/4960/sample/0521814960ws.pdf

Abstract Regular Polytopes

The concept of regular incidence complexes generalizes the notion of regular polyhedra in a combinatorial and grouptheoretical sense. A regular (incidence) complex K of dimension d (or briefly a d-complex) is a special type of partially ordered set with regularity defined by the flag-transitivity of its group A(K) of automorphisms.

Reguläre Inzidenzkomplexe III

Chirality and projective linear groups

The concept of regular incidence-complexes generalizes the notion of regular polyhedra in a combinatorial sense. A regular incidence-complex is a partially ordered set with regularity defined by certain transitivity properties of its automorphism group. The concept includes all regular d-polytopes and all regular complex d-polytopes as well as many geometries and well-known configurations.

Egon Schulte

Papers

Abstract Regular Polytopes

Reguläre Inzidenzkomplexe III

Chirality and projective linear groups

Regulre Inzidenzkomplexe III

A Polyhedral Realization of Felix Klein's Map {3, 7}8 on a Riemann Surface of Genus 3