1.1. CONGRUENCE RELATIONS. 3
we have s = t and, for every τ ∈ Irrf(Δ), there is a unique σ ∈ IrrF (Δ) such that
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) =
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
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
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.
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:
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
Such an isomorphism amounts to the set of equalities:
m1(σ)d(σ, τ) =