Semisimplicity and tensor products of group representations: Converse theorems. (With an appendix by Walter Feit)
Let \(k\) be a field of characteristic \(p\geq 0\) and \(G\) a group. A \(G\)-module is semisimple if it is a direct sum of simple modules. Let \(V\) and \(W\) be finite-dimensional \(G\)-modules. The following results are known: Theorem 1. If \(V\) and \(W\) are semisimple, then the tensor product \(V\otimes W\) is semisimple if \(p=0\), or if \(\dim V+\dim W<p+2\) (the second result is due to the author [see Invent. Math. 116, No. 1-3, 513-530 (1994; Zbl 0816.20014)]). Theorem 2. If \(V\) is semisimple, then the exterior square \(\Lambda^2V\) is semisimple if \(p=0\) or if \(p>0\) and \(\dim V\leq(p+3)/2\) (the paper above, Theorem 2). The author is interested in ``converse theorems'': proving the semisimplicity of \(V\) from that of \(V\otimes W\) or of \(\Lambda^2V\). The results are the following: Theorem 3. If \(V\otimes W\) is semisimple then \(p\) does not divide \(\dim W\). This result is best possible as an example shows. Theorem 4. If the tensor power \(\otimes^mV\) is semisimple then \(V\) is semisimple (\(n\in\mathbb{N}\)). Theorem 5. If \(\Lambda^2V\) is semisimple, then \(V\) is semisimple unless \(\dim V\equiv 2\pmod p\). Theorem 6. If the symmetric square \(\text{Sym}^2V\) is semisimple, then \(V\) is semisimple unless \(\dim V\equiv-2\pmod p\). Theorem 7. If the \(m\)th exterior power \(\Lambda^mV\) is semisimple, then \(V\) is semisimple unless \(\dim V\equiv 2,3,\dots,m\pmod p\). Examples show that the congruence conditions occurring in Theorems 5-7 cannot be suppressed. The first two examples are due to W. Feit (see the appendix). In all examples \(G\) is finite.
scientific article; zbMATH DE number 1066136
scientific article