The quasisimple case of the \(k(GV)\)-conjecture
Let \(F\) be a finite field of characteristic \(p\) and let \(G\) be a finite group of \(p'\)-order. Let \(V\) be a finite-dimensional faithful \(FG\)-module. Let \(v\) be any vector in \(V\) and let \(C_G(v)\) denote the subgroup of \(G\) that fixes \(v\). We say that \(v\) is an \(F\)-real vector if \(V\), considered as an \(FC_G(v)\)-module, is isomorphic to its dual module. We say that \(v\) is an \(F\)-weakly real vector if \(V\), considered as an \(FC_G(v)\)-module, contains a faithful submodule that is isomorphic to its dual. Interest in the existence of such real or weakly real vectors arises from their application to the solution of a special case of Brauer's \(k(B)\)-problem, concerning the number of irreducible complex characters in a \(p\)-block \(B\) of a finite \(p\)-solvable group. The author considers the problem of showing the existence of an \(F\)-real or \(F\)-weakly real vector when \(G\) contains a normal quasisimple subgroup \(E\) that acts irreducibly on \(V\). His main theorem is that in these circumstances an \(F\)-real vector exists except in three different cases, as follows. \(\bullet\) \(E\cong\text{SL}_2(5)\), \(\dim V=2\), \(|F|=11\), \(19\), or \(31\). \(\bullet\) \(E\cong 2.A_6\), \(\dim V=4\), \(|F|=7\). \(\bullet\) \(E\cong\text{Sp}_4(3)\), \(\dim V=4\), \(|F|=7\), \(13\), \(19\) or \(31\). In the third case, when \(|F|=31\), there is a weakly real vector for \(G\), but in all the other cases there is no weakly real vector. (Note that in general, \(G\) is strictly bigger than \(E\), and \(E\) may itself have real vectors.) The importance of this theorem is that, in combination with the work of several other researchers, it shows that Brauer's \(k(B)\)-conjecture is true for all \(p\)-solvable groups unless possibly \(p\) is one of \(3\), \(5\), \(7\), \(11\), \(13\), \(19\) or \(31\). The author builds on earlier work of \textit{D. P. M. Goodwin} [J. Algebra 227, No. 2, 395-432, 433-473 (2000; Zbl 0970.20005, Zbl 0970.20006)], who had left a finite list of potential exceptions to the existence of a real vector. Riese investigates these exceptional cases in greater detail, using the Atlas to provide character information and a computer calculation in one case.
scientific article; zbMATH DE number 1563171
scientific article