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Let $H$ be a semisimple Hopf algebra over an algebraically closed field$\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. We show thatthe antipode $S$ of $H$ satisfies the equality$S^2(h)=\mathbf{u}h\mathbf{u}^{-1}$, where $h\in H$,$\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$ and $\Lambda$ is a nonzero integralof $H$. The formula of $S^2$ enables us to define higher Frobenius-Schurindicators for the Hopf algebra $H$. This generalizes the notions of higherFrobenius-Schur indicators from the case of characteristic 0 to the case ofcharacteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. These indicators defined hereshare some properties with the ones defined over a field of characteristic 0.Especially, all these indicators are gauge invariants for the tensor categoryRep$(H)$ of finite dimensional representations of $H$.

Higher Frobenius-Schur indicators for semisimple Hopf algebras in positive characteristic