Representations of some Hopf algebras associated to the symmetric group S_n

Abstract

We study the representations and their Frobenius-Schur indicators of two semisimple Hopf algebras related to the symmetric group $S_n$, namely the bismash 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 characteristic 0. Both algebras are constructed using the standard representation of $S_n$ as a factorizable group, that is $S_n = S_{n-1}C_n = C_nS_{n-1}$. We prove that for $H_n$, the indicators of all simple modules are +1. For the dual Hopf algebra $J_n = k^{S_{n_1}}# kC_n$, the indicator can have values either 0 or 1. When $n = p$, a prime, we obtain a precise result as to which representations have indicator +1 and which ones have 0; in fact as $p \to \infty$, the proportion of simple modules with indicator 1 becomes arbitrarily small. We also prove a result about Frobenius-Schur indicators for more general bismash products $H =k^G# kF$, coming from any factorizable group of the form $L = FG$ such that $F\cong C_p.$ We use the definition of Frobenius-Schur indicators for Hopf algebras, as described in work of Linchenko and the second author, which extends the classical theorem of Frobenius and Schur, for a finite group $G.$