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
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
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
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
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,