1.1. CONGRUENCE RELATIONS. 3

we have s = t and, for every τ ∈ Irrf(Δ), there is a unique σ ∈ IrrF (Δ) such that

Lσ/mLσ

∼

=

Uτ . Furthermore, Pτ

∼

=

Lσ and w(X, τ) = λ(X, σ).

The values of the w(X, τ)’s can be determined from the representation space

XO/mXO for Δ over the field f. The action of Δ on this vector space may be

non-semisimple, but there is a unique, maximal semisimple quotient space. The

characterization of Pτ implies that w(X, τ) is equal to the multiplicity of τ in that

semisimple quotient. We will refer to w(X, τ) as the weight of τ in X. The structure

of X as a Zp[Δ]-module is completely determined by the w(X, τ)’s.

As an illustration, suppose that X is a free Zp[Δ]-module of rank 1. Of course,

X is then projective and one can show that w(X, τ) = n(τ) for each τ ∈ Irrf(Δ).

Thus, the Pτ ’s are direct summands in XO, which is a free O[Δ]-module of rank 1.

Each Pτ occurs with multiplicity n(τ). Note that if t 1, then Pτ itself will not be

free. If t = 1, then Δ is a p-group, and projective modules must be free. We will

return to this very special case below.

Now consider Pτ ⊗O F, a representation space for Δ over F. Each σ ∈ IrrF (Δ)

occurs with a certain multiplicity in Pτ ⊗O F. We denote this multiplicity by d(σ, τ).

We have VF

∼

= XO ⊗O F and the multiplicity λ(X, σ) of σ in the Δ-representation

space VF is obviously given by

(1.1.c) λ(X, σ) =

τ

d(σ, τ)w(X, τ)

where τ runs over Irrf(Δ). Thus, assuming that one can determine the d(σ, τ)’s,

formula (1.1.c) shows that the w(X, τ)’s determine the λ(X, σ)’s. The converse is

also true, as we will explain below.

Note that the quantities d(σ, τ) are purely group-theoretic in nature and do not

depend on X. For each σ ∈ IrrF (Δ), let Lσ denote any Δ-invariant O-lattice in Wσ.

Then Lσ/mLσ is a representation space for Δ over f. We denote the corresponding

representation by σ. It depends on the choice of Lσ, but its semisimplification σss

is determined up to isomorphism by σ. One of the basic results in modular rep-

resentation theory is that d(σ, τ), as defined above, coincides with the multiplicity

of τ in

σss.

That is, in a composition series for the O[Δ]-module Lσ/mLσ, the

number of composition factors isomorphic to Uτ is d(σ, τ). Later in this paper we

will use the notation

σss,τ

instead of d(σ, τ) to denote this multiplicity.

Suppose that ρ is any representation of Δ. We may assume that ρ is defined

over F. Just as above, we can realize ρ on a free O-module Lρ of rank n(ρ). The

reduction of ρ modulo m, which we denote by ρ, gives the action of Δ on Lρ/mLρ.

Its semisimplification

ρss

is uniquely determined by ρ and will be isomorphic to a

direct sum of the τ’s with certain multiplicities. Now suppose that we have two

representations ρi, i = 1, 2. For each i, ρi is isomorphic to a direct sum:

ρi

∼

=

σ

σmi(σ)

for certain multiplicities mi(σ), where σ varies over IrrF (Δ). Assuming that X

is a projective Zp[Δ]-module, a congruence relation arises whenever we have an

isomorphism

ρ1ss ∼

=

ρ2ss.

Such an isomorphism amounts to the set of equalities:

σ

m1(σ)d(σ, τ) =

σ

m2(σ)d(σ, τ)