5 Densest Packings 349full-dimensional). We say also thatΛis theorthogonal sumof the latticesΛ 1 andΛ 2.
The orthogonal sum of any finite number of lattices is defined similarly. A lattice is
indecomposableif it is not decomposable.
The following result was first proved by Eichler (1952).
Proposition 18Any latticeΛis an orthogonal sum of finitely many indecomposable
lattices, which are uniquely determined apart from order.
Proof (i) Define a vectorx∈Λto be ‘decomposable’ if there exist nonzero vectors
x 1 ,x 2 ∈Λsuch thatx=x 1 +x 2 and(x 1 ,x 2 )=0. We show first that every nonzero
x∈Λis a sum of finitely many indecomposable vectors.
By definition,xis either indecomposable or is the sum of two nonzero orthogonal
vectors inΛ. Both these vectors have square-norm less than the square-norm ofx,and
for each of them the same alternative presents itself. Continuing in this way, we must
eventually arrive at indecomposable vectors, since there are only finitely many vectors
inΛwith square-norm less than that ofx.
(ii) IfΛis the orthogonal sum offinitely many latticesLvthen, by the definition
of an orthogonal sum, every indecomposable vector ofΛlies in one of the sublat-
ticesLv. Hence if two indecomposable vectors are not orthogonal, they lie in the same
sublatticeLv.
(iii) Call two indecomposable vectorsx,x′‘equivalent’ if there exist indecompos-
able vectorsx=x 0 ,x 1 ,...,xk− 1 ,xk=x′such that(xj,xj+ 1 )=0for0≤j<k.
Clearly ‘equivalence’ is indeed an equivalence relation and thus the set of all indecom-
posable vectors is partitioned into equivalence classesCμ. Two vectors from different
equivalence classes are orthogonal and, ifΛis an orthogonal sum of latticesLvas in
(ii), then two vectors from the same equivalence class lie in the same sublatticeLv.
(iv) LetΛμbe the subgroup ofΛgenerated by the vectors in the equivalence class
Cμ. Then, by (i),Λis generated by the sublatticesΛμ. Since, by (iii),Λμis orthogo-
nal toΛμ′ifμ=μ′,Λis actually the orthogonal sum of the sublatticesΛμ.IfΛis
an orthogonal sum of latticesLvas in (ii), then eachΛμis contained in someLv.It
follows that eachΛμis indecomposable and that theseindecomposable sublattices are
uniquely determined apart from order.
LetΛbe a lattice inRn.IfΛ⊆Λ∗,i.e.if(x,y)∈Zfor allx,y∈Λ,thenΛis said
to beintegral.If(x,x)is an even integer for everyx∈Λ,thenΛis said to beeven.
(It follows that an even lattice is also integral.) IfΛis even and every vector inΛis an
integral linear combination of vectors inΛwith square-norm 2, thenΛis said to be a
root lattice.
Thus in a root lattice the minimal vectors have square-norm 2. It may be shown by a
long, but elementary, argument that any rootlattice has a basis of minimal vectors such
that every minimal vector is an integral linear combination of the basis vectors with
coefficients which are all nonnegative or all nonpositive. Such a basis will be called a
simplebasis. The facet vectors of a root lattice are precisely the minimal vectors, and
hence its Voronoi cell is the set of ally∈Rnsuch that(y,x)≤1 for every minimal
vectorx.
Any root lattice is an orthogonal sum of indecomposable root lattices. It was shown
by Witt (1941) that the indecomposable rootlattices can be completely enumerated;