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We study the representations and their Frobenius-Schur indicators of twosemisimple Hopf algebras related to the symmetric group $S_n$, namely thebismash products $H_n = k^{C_n}# kS_{n-1}$ and its dual $J_n = k^{S_{n-1}}#kC_n = (H_n)^*,$ where $k$ is an algebraically closed field of characteristic0. Both algebras are constructed using the standard representation of $S_n$ asa factorizable group, that is $S_n = S_{n-1}C_n = C_nS_{n-1}$. We prove thatfor $H_n$, the indicators of all simple modules are +1. For the dual Hopfalgebra $J_n = k^{S_{n_1}}# kC_n$, the indicator can have values either 0 or 1.When $n = p$, a prime, we obtain a precise result as to which representationshave indicator +1 and which ones have 0; in fact as $p \to \infty$, theproportion of simple modules with indicator 1 becomes arbitrarily small. Wealso prove a result about Frobenius-Schur indicators for more general bismashproducts $H =k^G# kF$, coming from any factorizable group of the form $L = FG$such that $F\cong C_p.$ We use the definition of Frobenius-Schur indicators forHopf algebras, as described in work of Linchenko and the second author, whichextends the classical theorem of Frobenius and Schur, for a finite group $G.$

Representations of Some Hopf Algebras Associated to the Symmetric Group S n