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 that the 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 integral of $H$. The formula of $S^2$ enables us to define higher Frobenius-Schur indicators for the Hopf algebra $H$. This generalizes the notions of higher Frobenius-Schur indicators from the case of characteristic 0 to the case of characteristic $p>\dim_{ \mathbbm{ k}}(H)^{ 1/2}$. These indicators defined here share some properties with the ones defined over a field of characteristic 0. Especially, all these indicators are gauge invariants for the tensor category Rep$(H)$ of finite dimensional representations of $H$.
Categories: math.RT