4 1. INTRODUCTION.
for all τ Irrf(Δ). Formula (1.1.c) then has the following consequence:
(1.1.d)
σ
m1(σ)λ(X, σ) =
σ
m2(σ)λ(X, σ) .
This is a nontrivial equation if ρ1

=
ρ2.
We call (1.1.d) a congruence relation because it arises from an isomorphism
ρ1ss

=
ρ2ss,
which we think of as a kind of congruence modulo m between the
two representations ρ1 and ρ2. Just as the conjugacy relation mentioned previously
arises whenever two representations are conjugate over Qp and something which we
call a duality relation (mentioned in section 1.3) arises whenever two representations
are dual to each other, a congruence relation arises whenever two representations
are congruent to each other in the above sense.
Such nontrivial congruence relations will obviously occur if t s. Indeed,
let us denote the Grothendieck group of finite-dimensional representations of Δ
over F by RF (Δ), which can be defined to be the free Z-module on IrrF (Δ). We
define
Rf(Δ) in the same way. One defines a homomorphism d : RF (Δ) Rf(Δ)
by sending the class [ρ] to the class [ρss]. An isomorphism
ρ1ss

=
ρ2ss
simply
means that [ρ1] [ρ2] ker(d). It is obvious that the Z-rank of ker(d) is at
least s t. The congruence relations that are described by (1.1.d) state that the
homomorphism λX : RF (Δ) Z defined by λX (σ) = λ(X, σ) for all σ IrrF (Δ)
factors through the map d. To be precise, define a homomorphism wX : Rf(Δ) Z
by wX (τ) = w(X, τ) for all τ Irrf(Δ). Then, λX = wX d. In essence, this is
just formula (1.1.c).
A theorem of Brauer asserts that d is surjective. (See [Se3], theorem 33.) It
follows that ker(d) has Z-rank equal to s t. Now RF (Δ) and Rf(Δ) are free
Z-modules with bases IrrF (Δ) and Irrf(Δ), respectively. Let Dp(Δ) denote the
matrix for d with respect to those bases, which we refer to as the decomposition
matrix for Δ and p. Indexing the rows of Dp(Δ) by Irrf(Δ) and the columns by
IrrF (Δ), it is a t × s matrix and d(σ, τ) is the entry on row τ, column σ. Since d is
surjective, it follows that Dp(Δ) has rank t. Hence one can use (1.1.c) for a certain
set of σ’s (of cardinality t) to determine, in principle, the values of the w(X, τ)’s.
Thus, all of the λ(X, σ)’s are then determined. A similar remark concerns the
parity of these invariants. Since the reduction of Dp(Δ) modulo 2 also has rank t,
one can determine the parity of λ(X, σ) for all σ’s if one knows that parity for a
suitable subset consisting of t of the σ’s.
The form of congruence relations depends on the group Δ. We will always
denote the trivial representations for Δ over F by σ0, and the trivial representation
over f by τ0, respectively. Of course, σ0 = τ0. As the first and simplest example,
suppose that Δ is a p-group. One can take F = Qp(μpa ), where pa is the maximal
order of elements in Δ. Then f = Fp, t = 1, and Irrf(Δ) = {τ0}. If σ IrrF (Δ),
then
σss

= τ0
n(σ)
and d(σ, τ0) = n(σ). If we assume that X is a projective Zp[Δ]-
module, then we have the congruence relation λ(X, σ) = n(σ)λ(X, σ0) for each
σ IrrF (Δ). However, in this case, it is not hard to show that Zp[Δ] is a local
ring and hence that any projective module X must be free. The above congruence
relation is then obvious.
Another relatively simple situation occurs if Δ is a p-solvable group, i.e., if Δ
has a composition series in which each simple subquotient is either of order p or
of order prime to p. According to the Fong-Swan theorem (theorem 38 in [Se3]),
every τ Irrf(Δ) is then of the form τ = σ for some σ IrrF (Δ). That σ may not
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