Let B = [b1, …,
bn] (with column vectors bi)
be a basis of ℝn. Then
L = [sum ]biℤ is a
lattice in ℝn and A = B[top ]B
is the Gram matrix of B. The reciprocal lattice L* of L has
basis B* = (B−1)[top ] with Gram matrix
A−1. For any nonsingular matrix
A = (ai,j) with inverse
A−1 = (a*i,j), let
τ(A) = max1[les ]i[les ]n
{[sum ]nj
=1[mid ]ai,j
·a*j,j[mid ]}. Then
τ(A), τ(A−1)[ges ]1 holds,
with equality for an orthogonal basis. We will show that for any lattice L there
is a basis with Gram matrix A such that
τ(A), τ(A−1)
= exp (O((ln n)2)). This generalizes a result in
[8] and [20].
For any basis transformation A→Ā with
Ā = T[top ]AT,
T = (ti,j)∈SLn(ℤ), we will show
[mid ]ti,j[mid ][les ]τ(A−1)
·τ(Ā). This implies that every integral matrix representation of a
finite group is equivalent to a representation where the coefficients of the matrices
representing group elements are bounded by exp (O((ln n)2)). This
new bound is considerably smaller than the known (exponential) bounds for automorphisms of
Minkowski-reduced lattice bases: see, for example, [6].
The quantities τ(A), τ(A−1)
have the following geometric interpretation. Let V(L) [ratio ]=
{x∈ℝn[mid ]∀λ∈L
[ratio ][mid ]x[mid ][les ][mid ]x−λ[mid ]} be the Voronoi cell
(also called the Dirichlet region) of the lattice L. For a basis B of
L, we call C(B) =
{[sum ]xibi,
[mid ]xi[mid ][les ]1/2} the basis cell of B. Both
cells define a lattice tiling of ℝn (see [6]); they
coincide for an orthogonal basis. For a general basis B of L with Gram
matrix A we will show
V(L)[les ]τ(A−1)·C(B)
and C(B)[les ]n·τ(A)·V(L).