Some Notes on Higher Frobenius–Schur Indicators of the Regular Representations for Matched Pairs of Groups

The notion of matched pair (F,G,▹,◃) of finite groups was introduced by Takeuchi in 1981, which is equivalent to a factorization of the group F⋈G. We find in this paper some sufficient conditions when equations νm(F⋈G)=νm(F)νm(G) for all m≥0 imply that F⋈G is the external direct product of F×G, where νm denotes m-th indicator of the regular representation of a finite group. A comparison with indicators of bismash product Hopf algebras CG#CF is also mentioned.