Irreducible tensor products of representations of finite quasi-simple groups of Lie type (Q2776266)

Let \(G\) be a finite group. We say that \(G\) has the property \({\mathcal P}(0)\) if, whenever \(U\) and \(V\) are irreducible complex \(G\)-modules, each of dimension greater than 1, the tensor product \(U\otimes V\) is a reducible \(G\)-module. Similarly, let \(r\) be a prime and let \(F\) be an algebraically closed field of characteristic \(r\). We say that \(G\) has the property \({\mathcal P}(r)\) if, whenever \(X\) and \(Y\) are irreducible \(FG\)-modules, each of dimension greater than 1, \(X\otimes Y\) is a reducible \(FG\)-module. The purpose of the paper under review is to determine whether or not \(G\) has the property \({\mathcal P}(0)\) or the property \({\mathcal P}(r)\) for any prime \(r\), when \(G\) is a quasi-simple group.NEWLINENEWLINENEWLINEThe authors briefly discuss the case that \(G\) is a simple sporadic group or a covering group of a sporadic group. Using the GAP system, they have shown that the groups \(M_{11}\), \(J_1\), \(2HS\), \(HN\), \(Ly\), \(He\), \(Fi_{23}\) and \(J_4\) have the property \({\mathcal P}(0)\), but the remaining sporadic groups do not have the property \({\mathcal P}(0)\). For example, when \(G\) is the Monster simple group \(M\), there are three non-trivial irreducible tensor products \(U\otimes V\), in one of which the factor \(U\) is the well known module of dimension 196883. Complete details of all the irreducible tensor products are not given.NEWLINENEWLINENEWLINEThe remainder of the paper is devoted to investigating the problem when \(G\) is a quasi-simple group of Lie type defined over the field \(\mathbb{F}_q\) of characteristic \(p\). In this case, the prime \(r\) is not allowed to equal \(p\), since Steinberg's twisted tensor product theorem guarantees that, when \(q\) is greater than \(p\), \(G\) will not have the property \({\mathcal P}(p)\).NEWLINENEWLINENEWLINEWe will briefly describe some of the authors' main results with respect to the property \({\mathcal P}(0)\). Suppose that \(G\) is as described above. Then either \(G\) has the property \({\mathcal P}(0)\) or else one of the following holds:NEWLINENEWLINENEWLINE(a) \(q\leq 3\). (b) \(G=\text{Sp}_{2n}(5)\). (c) \(G=\text{Sp}_{2n}(q)\), where \(q\) is a power of \(2\). (d) \(G/Z(G)=\text{PSL}_3(4)\).NEWLINENEWLINENEWLINEThe authors also show that \(\text{Sp}_{2n}(q)\) does not have the property \({\mathcal P}(0)\) when \(q=3\) or \(q=5\). The same is also true for \(\text{SU}_n(2)\) and \(\text{Sp}_{2n}(2)\) when \(n\geq 3\). With regard to groups which do not have the property \({\mathcal P}(r)\), there is a similar, larger list of possible candidates, but the authors conjecture that most of the possibilities do not occur. As is often the case in problems of this kind, extensive use is made of certain large parabolic subgroups in which the unipotent radical is a \(p\)-group of so-called special type. Such parabolic subgroups played a role in the proof of the Landazuri-Seitz-Zalesskii theorem and indeed the authors make use of some recent improvements in the bounds provided by the L-S-Z theorem.NEWLINENEWLINEFor the entire collection see [Zbl 0977.00027].