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
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.