4 W. PLESKEN AND G. NEBE
IL Notation, basic definitions, and constructions
For a rational finite matrix group G of degree n let
Z(G) := {M Qlxn  M full ^Gsublattice of(?1XTl}
and T0(G) := {F € F(G)  F positive definite }.
Note £(G ) ^ 0 and J^oJG) 7^ 0. For an (ndimensional) lattice L in Qlxn and a
positive definite symmetric rational matrix F of degree n, for short F ^Q^ymOf ^
Attt(F, L) := {g € GLn(Q) \ Lg = L and gFgtr = F}
be the automorphism group of F on Ly and more generally, for T C Q™ym,o ^
B(!F)L) :— f]Fejr Aut(Fy L) be the Bravais group of ^ on L. Finally, B(GiL) :=
B{3r o{G),L) for L e 2(G) denotes the Bravais group of G on L. If £ = ^ l x n , one
writes Au%(F),#(G ) etc. instead of 4u£(F, L),B(G, L) etc.
The following simple fact will be used quite often: G is a r.i.m.f. group if and only
if G is irreducible and G = Aut(F, L) for any pair (F, L) E To{G) x Z(G).
For (F,L) G ^"o(G) x Z(G) call
L
#(F)
: =

x G ( ?
nxi J
x F y
t r
€
%
f o r a U y € L

G
^ ^
the dua/ lattice of £ with respect to F . F is integral on £, if L*(F) D L, and F is
primitive on L, if £ # ( F ) 2 L and pL*^ 2 £ f° r a ^ primes p. If F is integral on
L, the isomorphism type of L^F^/L as abelian group is referred to as the elementary
divisors of F on L and \L*WjL\ =: det(F,L).
Note, if G is uniform, i.e. dimQ(T(G)) = 1, for every £ € ^(G ) there is exactly
one F € «Fo(G), which is primitive on £. If not specified otherwise, we will refer to
this form, if we talk about L (for instance we write £# instead of L#(F)).
If F is integral on £, one says F is even on £, if xFxtr € 2 ^ for all x G £, otherwise
F is odd on L. In the latter case
Lev^ := {rr € I I zF;r*r G 2 ^ } € Z(G)