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We introduce Brauer characters for representations of the bismash products of groups in characteristic p > 0, p not 2 and study their properties analogous to the classical case of finite groups. We then use our results to extend to bismash products a theorem of Thompson on lifting Frobenius-Schur indicators from characteristic p to characteristic 0

In this paper we show that a certain algebra being a comodule algebra over the Taft Hopf algebra of dimension $n^2$ is equivalent to a set of identities related to the $q$-binomial coefficient, when $q$ is a primitive $n^{th}$ root of 1. We then give a direct combinatorial proof of these identities.

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras., Comment: To Appear in Advances in Mathematics (2012), 23 pages

We derive a formula for the trace of the antipode on endomorphism algebras of simple self-dual modules of nilpotent liftings of quantum planes. We show that the trace is equal to the quantum dimension of the module up to a nonzero scalar depending on the simple module., Comment: 12 pages

In this paper we introduce a trace-like invariant for the irreducible representations of a finite dimensional complex Hopf algebra H. We do so by considering the trace of the map induced by the antipode S on the endomorphisms End(V) of a self-dual module V. We also compute the values of this trace for the representations of two non-semisimple Hopf algebras: u_q(sl_2) and D(H_n(q)), the Drinfeld double of the Taft algebra., Comment: 12 pages

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 $G.$