Computing with Arithmetic Groups

1 PAIRS OF DUAL CONES Around 1900 Voronoï [10] formulated his fundamental algorithm to enumerate all similarity classes of perfect lattices in a given dimension. This algorithm has far reaching generalisations ([7], [6], [4]) used to compute generators and relators for arithmetic groups. A quite general situation, where one may apply Voronoï’s algorithm, is described in [7]: Let σ : V1 ×V2 → R be a non-degenerate bilinear mapping on a pair of isomorphic finite-dimensional real vector spaces V1, V2. Two open non-empty subsets Pi ⊆ Vi form a pair of dual cones, if σ is strictly positive on P1 × P2 and for all f ∈ V1 \ P1, y ∈ V2 \ P2 there are f ′ ∈ P1, y′ ∈ P2 such that σ ( f ,y′) ≤ 0, σ ( f ′,y) ≤ 0. We now fix a discrete subset D ⊆ P2 \ {0}. For f ∈ P1 we define • The minimum of f as min( f ) := min{σ ( f ,d ) | d ∈ D}, • S ( f ) := {d ∈ D | σ ( f ,d ) = min( f )}, • and the Voronoï domain V ( f ) := {∑d ∈S (f ) add | ad ≥ 0}. • The element f is called perfect, if S ( f ) spans V2, so if V ( f ) has a non-empty interior. • PD := { f ∈ P1 | min( f ) = 1, f is perfect } denotes the set of perfect elements of minimum 1. In his original application Voronoï aimed to classify all locally densest lattices. Here