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In-Depth Information
Lemma 7.2.5.2 and some purely algebraic manipulations of matrices lead to
7.2.5.3. Theorem.
It is possible to choose bases for all the groups C
q
(K) simul-
taneously with respect to which the qth incidence matrix has the normalized
form
n
columns columns
64467
-
g
g
q
+1
0
q
44
q
44
8
q
Ê
0
d
0
ˆ
¸
1
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
˜
˜
˜
˜
˜
˜
˜
˜
˜
˜
Ô
q
d
2
g
rows
˝
q
O
Ô
Ô
¸
q
N
=
*
q
0
0
0
d
˛
(7.3)
g
q
0
0
0
0
Ô
n
-
g
rows
˝
q
q
Ô
Ô
˛
Ë
¯
0
0
0
0
where
(1) the d
q
's are positive integers,
(2) d
i+1
divides d
i
, and
(3) n
q
-g
q
≥g
q-1
.
Outline of Proof.
Choose a basis
{}
=
{
}
q
qq
q
c
c
,
c
,...,
c
12
n
q
for each group C
q
(K) and assume that N
q
is the qth incidence matrix with respect
to these bases. We shall transform the matrices N
q
into the form (7.3) by appropri-
ate changes to the bases and so we need to know how changes to bases affect the
matrices. First of all, note that changing the basis {c
q
} clearly affects both N
q
and
N
q-1
.
The matrices N
q
have the following properties:
Claim 1.
(a) Replacing
{
}
{
}
q
q
q
q
q
q
c
,...,
c
,...,
c
by
c
,...,
-
c
,...,
c
1
i
1
i
n
n
q
q
corresponds to changing all signs in the ith column of N
q-1
and ith row of N
q
.
(b) Replacing
{
}
{
}
q
q
q
q
q
q
q
q
c
,...,
c
,...,
c
,...,
c
by
c
,...,
c
,...,
c
,...,
c
1
i
j
1
j
i
n
n
q
q
