# Composition factors of tensor products of modular representations

In ordinary representation theory over $\mathbb{C}$, all the irreducible modules of a finite group $G$ appear as composition factors of the tensor products $X \otimes \cdots \otimes X$ of a faithful $\mathbb{C}$-representation $X$. This is e.g. Thm. 10.8 in Ch.V of Huppert's Book "Finite Groups I".

This should also be true in positive characteristic, so let $X:G \to \mathrm{GL}(n,K)$ with $\mathrm{char}(K) = p$ be a modular faithful representation. Then all irreducible $KG$-modules appear as composition factors of the tensor products $X \otimes \cdots \otimes X$.

I'm not sure, how to prove this, but I think it shouldn't be difficult.

This is true. I believe it may be due to L.G. Kovacs. When $K$ is algebraically closed, the proof is much the same as the complex case, except that one works with the Brauer character $\phi$ afforded by $X.$ For any Brauer irreducible character $\psi,$ let $\theta$ denote the Brauer character of its projective cover. Then some linear combination of powers of $\phi$ vanishes on all non-identity $p$-regular elements, but not on $1_{G}.$ Hence $\langle \theta, \phi^{n} \rangle \neq 0$ for some $n,$ so that the simple module affording $\psi$ is a composition factor of $X \otimes \ldots \otimes X$ ( $n$ factors) The case when $K$ is a finite field follows from the algebraically closed case.
As Geoff points out, it's natural here to start with $K$ being algebraically closed; so irreducible means the same thing as absolutely irreducible. In any case, the contrasting methods of Steinberg and Brauer (note his comments) in their two short notes provide lots of information about what is possible in prime characteristic. There is no need to start from scratch here. Both of those articles are available online here and here.
• Thank you for the references. I was only interested in the case, when $K$ is algebraically closed. – KoopaTroopa Oct 5 '13 at 19:53