Higher indicators for some groups and their doubles. (Q2885391)

The higher Frobenius-Schur indicators of a semisimple Hopf algebra were introduced by \textit{Y. Kashina, Y. Sommerhäusser} and \textit{Y. Zhu} [in Mem. Am. Math. Soc. 855 (2006; Zbl 1163.16029)], extending the previous definition of the second indicator given by \textit{V. Linchenko} and \textit{S. Montgomery} [Algebr. Represent. Theory 3, No. 4, 347-355 (2000; Zbl 0971.16018)].NEWLINENEWLINE In the paper under review the author considers a finite group \(G\) such that \(G\) is a semidirect product of two cyclic groups \(G=\mathbb Z_k\rtimes\mathbb Z_{ql}\) with respect to an action induced by a group automorphism of prime order \(q\), including the dihedral and semidihedral groups and nonabelian groups of order \(pq\), where \(p\neq q\) are prime numbers. For such a group \(G\), he determines the higher indicators of all irreducible representations of \(G\) as well as those of the Drinfel'd double \(D(G)\). This is also done in the case where \(G\) is a generalized quaternion group. The values of the indicators in both families of examples considered in this paper turn out to be integers. An irregular nilpotent group of order \(5^6\) with non-integer indicators for the Drinfel'd double has recently been found by \textit{M. Iovanov, G. Mason} and \textit{S. Montgomery} [\(FSZ\)-groups and Frobenius-Schur indicators of quantum doubles, preprint \url{arXiv:1208.4153} (2012)].