The Loewy length of a tensor product of modules of a dihedral two-group. (Q392503)

Let \(q\geq 2\) be a \(2\)-power, let \(D_{4q}\) denote the dihedral group of order \(4q\) and let \(k\) be a field of characteristic \(2\). The group algebra \(kD_{4q}\) has tame representation type; the indecomposable \(kD_{4q}\)-modules were classified by \textit{C. M. Ringel} [Math. Ann. 214, 19-34 (1975; Zbl 0299.20005)].NEWLINENEWLINE The main result of the paper under review determines the Loewy length of the tensor product \(A\otimes B\) of any two indecomposable \(kD_{4q}\)-modules \(A\) and \(B\) such that both \(A\) and \(B\) have simple top and socle.NEWLINENEWLINE The authors show that the Loewy length of a tensor product \(M\otimes N\) (of modules for any finite group) is the maximum of the Loewy lengths of the modules \(A\otimes B\) where \(A\) is a subquotient of \(M\) and \(B\) is a subquotient of \(N\), such that both \(A\) and \(B\) have simple top and socle. The maximum Loewy length of an indecomposable \(kD_{4q}\)-module is \(2q+1\), and it is clear that this is attained only for the regular module. Thus a corollary of the main result is a complete classification of all \(kD_{4q}\)-modules \(M\) and \(N\) such that \(M\otimes N\) has a projective summand.