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PRIMITIVE Z2 ⊕ Z2-MANIFOLDS 215
clear that all of them are decomposable or all of them are indecomposable
simultaneously. By the representation defined by A and B we will refer to
the representation ρ defined by
ρ(1, 0) = A and ρ(0, 1) = B.
In addition to (2.4) we assume that the representation defined by A and
B is indecomposable and has no fixed points.
To achieve the result we will go through the following steps.
(1) Show that A and B have some special canonical type (see Lemma 2.8).
(2) Show that conjugating, in Gl(n, Z), the special type of (1) is equivalent
to performing some elementary operations on rows and columns (see
Lemma 2.9).
(3) Reduce matrices A and B, according to (2), until it is clear whether
the representation decomposes or not.
From (2.1) and the assumption n ≥ 3 it follows that A = B and A = −B.
Otherwise, the representation defined by A and B decomposes. Observe
that, in particular, the representation is faithful.
It is not difficult to show that for any integral matrix A, of rank n, the
sub-lattice Λ = Ker A of Z n ⊆ R n admits a direct complement Λ c . That is,
always there exists Λ c , such that Z n = Λ ⊕ Λ c .
Let Λ = Ker(A − B). We have 0 ⊆ Λ ⊆ Z n and we can write
A =
A11 A12
A21 A22
, B =
B11 B12
B21 B22
Since A − B|Λ = 0, then A11 = B11 and A21 = B21. Moreover, from the
equation (A − B)(A + B) = A 2 − B 2 = 0 we get 2A21 = 0 and A22 = −B22.
Hence, we may assume that A and B are of the form
A =
A11 A12
A22
, B =
A11 B12
−A22
The identity A2 = I implies that A2 11 = I = A222 . Therefore, both
decompose as a direct sum of three blocks, an identity and a minus identity
block and a matrix K, being K a direct sum of matrices J (see (2.1)).
Moreover, the condition of having no fixed points, for the representation
defined by A and B, forces A11 to be −I. Finally, we get the following form
for A and B,
(2.5)
A =
where A3K = A3 and B3K = −B3.
.
.
−I A1 0
A3
−I 0 B2 B3
I
−I , B = −I ,
I
K
−K
Remark. Not all three blocks I, −I and K must be present in the decomposition
of A22.