Modular representations that look ordinary

Let G be a finite group and let (K, \mathcal{O}, k) be a p-modular system for G. Thus K is a field of characteristic zero sufficiently large that if V is any irreducible representation of G then \mathrm{End}_{KG}(V) \cong K and K contains all the eigenvalues of the matrices representing elements of G in their action on V, \mathcal{O} is an integrally closed subring of K containing all these eigenvalues, and k is a field of prime characteristic p obtained by quotienting \mathcal{O} by a maximal ideal \mathfrak{m}, necessarily lying above \langle p \rangle \unlhd \mathcal{O}. By localizing if necessary, we may assume that all elements of \mathcal{O} not in \mathfrak{m} are invertible. So \mathcal{O} is a local ring with unique maximal ideal \mathfrak{m}.

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 K, \mathcal{O} or k.

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

Proposition 1. Let \chi be an irreducible character of G such that \chi(1) is divisible by the highest power of p dividing |G|. If g \in G is a p-element then \chi(g) = 0.

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

e_\chi = \frac{\chi(1)}{|G|} \sum_{g \in G} \chi(g^{-1}) g.

Note that the coefficients of e_\chi are contained in \mathcal{O}.

Proof of Proposition 1. Let P be a Sylow p-subgroup of G containing g. Since

(1) \qquad \mathcal{O}G =  e_\chi \mathcal{O}G \oplus (1-e_\chi) \mathcal{O}G,

considered as a right \mathcal{O}G-module, e_\chi \mathcal{O}G is projective. Hence the restriction of e_\chi \mathcal{O}G to \mathcal{O}P is also projective. Now \mathcal{O}P is indecomposable as a right \mathcal{O}P-module, since any direct sum decomposition induces a direct sum decomposition of the indecomposable kP-module kP by reduction modulo \mathfrak{m}. Hence the unique projective indecomposable \mathcal{O}P-module is \mathcal{O}P itself and e_\chi \mathcal{O}G\downarrow_P \cong \mathcal{O}P \oplus \cdots \oplus \mathcal{O}P. Restricting further we get

(2)\qquad e_\chi \mathcal{O}G \downarrow_{\langle g \rangle} \cong \mathcal{O} \langle g \rangle^{\oplus \chi(1)/\mathrm{ord}(g)}.

From the canonical basis for the right-hand we see that the character of g acting on e_\chi \mathcal{O}G is zero. But, by Wedderburn’s Theorem, e_\chi KG is isomorphic to the direct sum of \chi(1) copies of a simple KG-module with character \chi. Hence \chi(1)\chi(g) = 0.\qquad\Box

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: \chi(g) = 0 whenever g has order divisible by p.

Outline proof. let g have order p^a c where c is not divisible by p, let g_p = g^c, and let g_{p'} = g^{p^a}. Take a direct sum decomposition of e_\chi KG into eigenspaces for the action of g_{p'}. By (2), g_p has trace zero in its action on each eigenspace, hence so does g.

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

Theorem 2. Let \chi be an irreducible character of G such that \chi(1) is divisible by the highest power of p dividing |G|. There is a projective \mathcal{O}G-module M with ordinary character \chi such that M/\mathfrak{m}M is a simple projective kG-module.

Proof. Let M be a \mathcal{O}G-module such that M_K = M \otimes_\mathcal{O} K is a simple KG-module. By Wedderburn’s Theorem, e_\chi KG is isomorphic to \mathrm{End}_K(M_K), thought of as the complete matrix algebra of \chi(1) \times \chi(1) matrices with coefficients in K. Let \rho : KG \rightarrow \mathrm{End}_K(M_K) be the corresponding ring homomorphism, with kernel (1-e_\chi)KG. The restriction of \rho to \mathcal{O}G has kernel (1-e_\chi)\mathcal{O}G. So it is sufficient to prove that the image of \rho : \mathcal{O}G \rightarrow \mathrm{End}_K(M_K) is \mathrm{End}_O(M), thought of as the complete matrix ring of \chi(1) \times \chi(1) matrices with coefficients in \mathcal{O}. Since M is a \mathcal{O}G-module, the matrix coefficients lie in \mathcal{O}.

The surjectivity of the restriction of \rho follows from the following ‘averaging’ formula which expresses an arbitrary \phi \in \mathrm{End}_\mathcal{O}(M) as something in its image:

\phi = \frac{\chi(1)}{|G|} \sum_{g \in G} \mathrm{Tr}_M\bigl( \rho(g^{-1})  \phi\bigr) \rho(g).

Subproof. Since the ring homomorphism \rho: KG \rightarrow \mathrm{End}_K(M) is surjective, it is sufficient to prove this when \phi = \rho(x) with x \in G; the right-hand side is

\rho(x) \frac{\chi(1)}{|G|} \sum_{g\in G}  \chi(g^{-1} x) \rho(x^{-1} g) = \rho(x) \rho(e_\chi) = \rho(x),

as required.

We can now think of the right \mathcal{O}G-module M very concretely, as those d \times d matrices with coefficients in \mathcal{O} that are zero outside their first row. Reducing mod \mathfrak{m}, we obtain M/\mathfrak{m}M; this is the unique simple module (up to isomorphism) for \mathrm{Mat}_{d \times d}(k). Hence M/\mathfrak{m}M is a simple projective kG-module, as claimed. \qquad\Box

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