Let be a finite group and let be a -modular system for . Thus is a field of characteristic zero sufficiently large that if is any irreducible representation of then and contains all the eigenvalues of the matrices representing elements of in their action on , is an integrally closed subring of containing all these eigenvalues, and is a field of prime characteristic obtained by quotienting by a maximal ideal , necessarily lying above . By localizing if necessary, we may assume that all elements of not in are invertible. So is a local ring with unique maximal ideal .

This is a standard setup for modular representation theory. The main purpose of this proof is to prove Theorem 2 below, which gives a precise description of a family of representations that look substantially the same, whether defined over , or .

As a warm-up we prove the following result on ordinary characters.

**Proposition 1.** *Let be an irreducible character of such that is divisible by the highest power of dividing . If is a -element then .*

We need the following fact: the central primitive idempotent in corresponding to is

Note that the coefficients of are contained in .

*Proof of Proposition 1.* Let be a Sylow -subgroup of containing . Since

considered as a right -module, is projective. Hence the restriction of to is also projective. Now is indecomposable as a right -module, since any direct sum decomposition induces a direct sum decomposition of the indecomposable -module by reduction modulo . Hence the unique projective indecomposable -module is itself and . Restricting further we get

From the canonical basis for the right-hand we see that the character of acting on is zero. But, by Wedderburn’s Theorem, is isomorphic to the direct sum of copies of a simple -module with character . Hence .

I find it quite striking that the modular theory gives a short proof of a result stated entirely in the language of ordinary (i.e. characteristic zero) representation theory. Moreover, from (2), we can easily get a stronger result: whenever has order divisible by .

*Outline proof.* let have order where is not divisible by , let , and let . Take a direct sum decomposition of into eigenspaces for the action of . By (2), has trace zero in its action on each eigenspace, hence so does .

Proposition 1 is also a corollary of the following theorem. My first version of the proof did not make it entirely clear that is a direct sum of isomorphic -modules, each simple as a -module. To repair the gap, I borrowed the proof of (1) (2) on page 162 of this recent book by Peter Webb.

**Theorem 2.** *Let be an irreducible character of such that is divisible by the highest power of dividing . There is a projective -module with ordinary character such that is a simple projective -module.*

*Proof.* Let be a -module such that is a simple -module. By Wedderburn’s Theorem, is isomorphic to , thought of as the complete matrix algebra of matrices with coefficients in . Let be the corresponding ring homomorphism, with kernel . The restriction of to has kernel . So it is sufficient to prove that the image of is , thought of as the complete matrix ring of matrices with coefficients in . Since is a -module, the matrix coefficients lie in .

The surjectivity of the restriction of follows from the following ‘averaging’ formula which expresses an arbitrary as something in its image:

*Subproof.* Since the ring homomorphism is surjective, it is sufficient to prove this when with ; the right-hand side is

as required.

We can now think of the right -module very concretely, as those matrices with coefficients in that are zero outside their first row. Reducing mod , we obtain ; this is the unique simple module (up to isomorphism) for . Hence is a simple projective -module, as claimed.