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