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This concerns one of those "well known" facts, referred to in a recent preprint I've been looking at. In principle it's elementary, but I can't pin down an explicit textbook reference for it. Start with two finite groups $A,B$ and their product $G:=A \times B$, working over a splitting field $K$ for the groups involved with prime characteristic dividing $|G|$. Let $S_1, \dots, S_m$ and $T_1, \dots, T_n$ be respective sets of representatives of isomorphism classes of simple modules for the group algebras $KA, KB$. In turn let the projective covers (=injective hulls) be respectively $P_i, Q_j$. These are the PIMs or indecomposable projective modules for the two group algebras.

It's a standard observation (found in some books) that there is an obvious isomorphism between $KG$ and the tensor product algebra $KA \otimes_K KB$, while each group algebra splits into the direct sum (as a left module over itself) of the various PIMs taken with multiplicity equal to the dimension of the corresponding simple module. It's also a standard fact (found in some books) that each $S_i \otimes T_j$ is a simple module for $KG$. From these ingredients one can conclude that $P_i \otimes Q_j$ is the corresponding PIM, thereby exhausting all isomorphism classes for $KG$.

Is all of this written down in a self-contained way somewhere?

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The paper Representations of direct products of finite groups. Burton Fein Source: Pacific J. Math. Volume 20, Number 1 (1967), 45-58.

Link

has what you are looking for and also explains what happens in the case that $K$ is not a splitting field. Look at Theorem 2.2 and the remark following it.

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  • $\begingroup$ Thanks very much for pointing out this old journal reference, which doesn't seem to be reflected in the later textbook literature (though I'm tempted to search further in the two large volumes by Curtis and Reiner Methods of Representation Theory, since Curtis was Burton Fein's thesis adviser at Oregon). $\endgroup$
    – Jim Humphreys
    Commented Oct 28, 2011 at 0:39
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I'm not sure what Fein does, but can't you do everything with Brauer characters, at least in the splitting field case. Use the fact that the Brauer characters of the PIMs are the unique class funcstion $\theta_i$ such that $\langle \theta_i, \phi_j \rangle = \delta_{ij}$where the $\phi_j$ are the Brauer characters of the simple modules. Since the simple modules of a direct product are easily determined, this uniquely determines the Brauer characters of the PIMs of the same direct product, and it's clear that they are the pairwise products of the Brauer characters of the PIMs for the two direct factors.

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  • $\begingroup$ Fein's paper (which has open access) is mainly concerned with working over general fields, adapting Schur index ideas, etc. The case of a splitting field seems fairly straightforward in his module language, but I couldn't find a textbook reference. Use of Brauer characters looks reasonable, though for uniqueness you have to specify that $\theta_i$ is the Brauer character of a projective module. The method I sketched based on dimension comparisons would be fairly direct too if filled in. $\endgroup$
    – Jim Humphreys
    Commented Nov 29, 2011 at 15:24
  • $\begingroup$ Well, I think of a Brauer character as defined only on $p$-regular elements, and the "inner product" of Brauer characters as restricted to $p$-regular elements. With this convention, the Brauer characters of the PIMs are uniquely determined once the Brauer characters of the simple modules are. $\endgroup$
    – Geoff Robinson
    Commented Dec 4, 2011 at 20:33

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