2 1. INTRODUCTION.

more irreducible representations for Δ in characteristic 0 than in characteristic p

when p |Δ|. To be precise, the cardinality of IrrF (Δ) is equal to the number of

conjugacy classes in the group Δ, which we denote by s. Let O denote the ring of

integers of F, m denote the maximal ideal of O, and f denote the residue field O/m,

a finite extension of the prime field Fp. Let Irrf(Δ) denote the set of irreducible

representations of Δ over f. These representations are absolutely irreducible because

of the choice of F. The cardinality of Irrf(Δ), which we denote by t, is equal to

the number of conjugacy classes in Δ of elements whose order is not divisible by p.

And so, obviously, t ≤ s. This inequality is strict if p |Δ|.

We will use the notation IrrF (G) and Irrf(G) throughout this paper, where G

is a finite group. Irreducible representations are always assumed to be absolutely

irreducible, unless otherwise mentioned, and it is implicit in the above notation

that F is a finite extension of Qp and is suﬃciently large so that all irreducible

representations of G in characteristic 0 are realizable over F and all irreducible

representations in characteristic p are realizable over its residue field f. As men-

tioned earlier, it suﬃces to have the roots of unity of a certain order in F. We

prefer F to be a finite extension because sometimes it is useful for the ring O

to be compact and Noetherian. This notation is also a simple way of indicating

whether the representations being considered are over a field of characteristic 0 or

of characteristic p.

1.1. Congruence relations.

Let XO = X⊗Zp O. We can view XO as an O[Δ]-module. If we assume that |Δ|

is not divisible by p, then formula (1.0.a) is reflected in the following decomposition

of XO:

(1.1.a) XO

∼

=

σ

Lσ(X,σ)λ

where Lσ is a Δ-invariant O-lattice in Wσ. Note that XO and each of the Lσ’s are

projective O[Δ]-modules. The isomorphism class of Lσ is uniquely determined by

σ.

The situation is not as simple if p divides |Δ|. However, under the assumption

that X is projective, there is a decomposition of XO which can be viewed as a

natural generalization of (1.1.a). For each τ ∈ Irrf(Δ), let Uτ denote the underlying

f-representation space for τ and let n(τ) = dimf(Uτ ). We can view Uτ as a simple

O[Δ]-module.

There is a projective O[Δ]-module Pτ which is characterized (up to isomor-

phism) as follows: Pτ has a unique maximal O[Δ]-submodule and the correspond-

ing quotient module is isomorphic to Uτ . One often refers to Pτ as the projective

hull of Uτ as an O[Δ]-module.

The Pτ ’s are precisely the indecomposable, projective O[Δ]-modules. Now

suppose that X is a projective Zp[Δ]-module. Then XO will be a projective O[Δ]-

module and we will have a decomposition

(1.1.b) XO

∼

=

τ

Pτ

w(X,τ)

where τ varies over Irrf(Δ) and w(X, τ) ≥ 0. This decomposition coincides with

that in (1.1.a) if we assume that |Δ| is not divisible by p. Under that assumption,