Automorphism groups and normal forms in Normaliz

TL;DR

This paper explores methods for computing automorphism groups and normal forms of cones and polyhedra within Normaliz, utilizing nauty, and demonstrates applications on complex geometric structures like the icosahedron.

Contribution

It introduces a comprehensive approach to automorphism computation in Normaliz, covering various automorphism types and implementation details with practical examples.

Findings

Automorphism groups for the icosahedron and linear ordering polytopes are determined.

Implementation of automorphism computation via nauty in Normaliz.

Different types of automorphisms, including algebraic, are effectively handled.

Abstract

We discuss the computation of automorphism groups and normal forms of cones and polyhedra in Normaliz, and indicate its implementation via nauty. The types of automorphisms include integral, rational, Euclidean and combinatorial, as well as algebraic for polytopes defined over real algebraic number fields. Examples treated in detail are the icosahedron and linear ordering polytopes whose Euclidean automorphism groups are determined.