Modular representations and indicators for bismash products. (Q371413)

In the paper under review, the authors concentrate on the representations of the bismash products \(H_{\Bbbk}=\Bbbk^G\#\Bbbk F\) associated to a group \(Q=FG\) factorizable into finite subgroups \(F,G\subset Q\) over an algebraically closed field \(\Bbbk\) of positive characteristic \(p\neq 2\). In the first part of the paper, the authors define the Brauer characters for the bismash products \(H_{\Bbbk}\) and prove that they have a lot of properties analogous to those for finite groups in positive characteristic. In particular, the authors extend to the bismash products \(H_{\Bbbk}\) the Brauer theorem on the determinant of the Cartan matrix. In the second part of the paper, the authors extend Thompson's theorem [\textit{J. G. Thompson}, Lect. Notes Math. 1185, 210-230 (1986; Zbl 0582.12006)] on Frobenius-Schur indicators for representations of finite groups to the case of the bismash products \(H_{\Bbbk}\).