\begin{equation}gin{document}

\title[Higher Frobenius-Schur indicators]{Higher Frobenius-Schur indicators for semisimple Hopf algebras in positive characteristic}

\author{Zhihua Wang}
\address{Z. Wang\newline Department of Mathematics, Taizhou University,
Taizhou 225300, China}
\email{mailzhihua@126.com}
\author{Gongxiang Liu}
\address{G. Liu\newline Department of Mathematics, Nanjing University,
Nanjing 210093, China}
\email{gxliu@nju.edu.cn}
\author{Libin Li}
\address{L. Li\newline School of Mathematical Science, Yangzhou University, Yangzhou 225002, China}
\email{lbli@yzu.edu.cn}
\date{}
\subjclass[2010]{16T05}
\keywords{semisimple Hopf algebra, $S^2$-formula, Frobenius-Schur indicator, gauge invariant}

\begin{equation}gin{abstract} 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 $H$.
\end{abstract}
\maketitle
\section{\bf Introduction}
Linchenko-Montgomery \cite{LM} generalized the classical Frobenius-Schur (FS) indicators from group-theoretic result to the setting of a semisimple involutory Hopf algebra $H$.
They also defined higher FS indicators $\nu_n(V)$ by using idempotent integral $\Lambda$ of $H$, namely, \begin{equation}gin{equation}\label{eq}\nu_n(V)=\chi_V(\Lambda_{(1)}\cdots\Lambda_{(n)})\ \text{for}\ n\geq1,\end{equation} where $\chi_V$ is the character afforded by finite dimensional representation $V$ of $H$. The higher FS indicators were later extensively studied by Kashina-Sommerh$\ddot{a}$user-Zhu for semisimple Hopf algebras over an algebraically closed field of characteristic zero \cite{KSZ}, and by Ng-Schauenburg for semisimple quasi-Hopf algebras over the field of complex numbers \cite{NS1}. The notions of higher FS indicators
have been generalized to objects of a pivotal category \cite{NS2, NS3}.

However, the notions of  higher FS indicators for semisimple Hopf algebras over a field of positive characteristic seem not to be considered (except for those semisimple involutory Hopf algebras).
In this paper, we consider higher FS indicators for a finite dimensional semisimple Hopf algebra $H$ over an algebraically closed field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. We need to point out that the Hopf algebra $H$ here is not necessarily involutory unless the characteristic $p$ is larger than a certain number (see \cite{So,EG}).

For the  antipode $S$ of $H$, we first obtain a formula for $S^2$ as follows:
$$S^2(h)=\mathbf{u}h\mathbf{u}^{-1},$$
where $h\in H$, $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$ and $\Lambda$ is a nonzero integral of $H$.

According to the formula of $S^2$, we have an isomorphism of $H$-modules $$j_{\mathbf{u},V}:V\rightarrow V^{**},\ \ j_{\mathbf{u},V}(v)(f)=f(\mathbf{u}\cdot v)\ \text{for}\ v\in V, f\in V^*,$$
which is functorial in $V$. As the element $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$ is not necessarily a group-like element, the functorial isomorphism $j_{\mathbf{u}}:id\rightarrow(-)^{**}$ is not necessarily a tensor isomorphism. In other words, the category Rep$(H)$ of finite dimensional representations of $H$ is not necessarily  pivotal with respect to the structure $j_{\mathbf{u}}$. Even though, using the functorial isomorphism $j_{\mathbf{u}}$ we may still define the $n$-th FS indicator $\nu_n(V)$ of $V$ to be the trace of a certain $\mathbbm{k}$-linear operator as Ng-Schauenburg did in \cite{NS2}. It is similar to the case of characteristic 0 that
the $n$-th FS indicator $\nu_n(V)$ defined here can also be entirely described in terms of the integral $\Lambda$ of $H$ and the character $\chi_V$ of $H$-module $V$:
\begin{equation}gin{equation}\label{eqq}\nu_n(V)=\chi_V(\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)})\ \text{for}\ n\geq1.\end{equation}
Moreover, the formula (\ref{eqq}) does not depend on the choice of
the nonzero integral $\Lambda$ and it recovers the original formula (\ref{eq}) when the characteristic of $\mathbbm{k}$ is zero and $\Lambda$ is idempotent.

Note that the formula (\ref{eqq}) can be written as $\nu_n(V)=\chi_V(\mathbf{u}^{-1}P_n(\Lambda))$ for $n\geq1$, where $P_{n}$ is the $n$-th Sweedler power map of $H$. Clearly, the $n$-th Sweedler power map $P_n$ is valid for all $n\in\mathbb{Z}$, this motivates us to extend the $n$-th FS indicator from $n\geq1$ to $n\in\mathbb{Z}$. That is, by definition, $\nu_n(V)=\chi_V(\mathbf{u}^{-1}P_n(\Lambda))$ for all $n\in\mathbb{Z}$. We find that the higher FS indicators defined over a field of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$ and the ones defined over a field of characteristic 0 share some common properties.
For instance, it is similar to the case of characteristic 0 (see \cite{KMN, KSZ}) that by replacing $V$ with the regular representation $H$, we reconstruct the $n$-th indicator of $H$, a notion defined by the trace of the map $S\circ P_{n-1}$. Also, it is similar to characteristic 0 case that $V$ and its dual $V^*$ have the same higher FS indicators. Especially, similar to the case of characteristic 0 that the $n$-th FS indicator $\nu_n(V)$ defined here is an invariant of the tensor category Rep$(H)$ for any $n\in\mathbb{Z}$ and any finite dimensional representation $V$ of $H$.

The paper is organized as follows: In Section 2, we present some basic results on semisimple Hopf algebras. In Section 3, we deduce the formula of $S^2$ by comparing two different forms of the character $\chi_H$ of the regular representation $H$.
We investigate some properties of the element $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$  and show that the integral $\Lambda$ of $H$ is cocommutative if and only if $S^2=id$.
In Section 4, we generalize the notions of higher FS indicators from characteristic 0 case to characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$ case and find that the indicators defined here share some  common properties with the ones defined over a field of characteristic 0. In Section 5, we show that the $n$-th FS indicator $\nu_n(V)$ is a  gauge invariant for any integer $n$ and any finite dimensional representation $V$ of $H$.

\section{\bf Preliminaries}
Throughout this paper, $H$ is a finite dimensional semisimple Hopf algebra over an algebraically closed field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. We need to stress that all results presented here are also valid for the case of characteristic 0, although we only deal with the case of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$.

As a Hopf algebra, $H$ has a counit $\varepsilon$, antipode $S$, multiplication $m$ and comultiplication $\Delta$. The comultiplication $\Delta(a)$ will be written as $\Delta(a)=a_{(1)}\otimes a_{(2)}$ for $a\in H$, where we omit the summation sign. We denote by $\Lambda$ and $\lambda$ the left and right integrals of $H$ and $H^*$ respectively so that $\lambda(\Lambda)=1$. Since the semisimple Hopf algebra $H$ is unimodular, the left and right integrals of $H$ are the same.
We refer to \cite{Mon} for basic theory of Hopf algebras.

If $V$ is a finite dimensional $H$-module, then $V$ is also called a representation of $H$ via the algebra homomorphism $\rho_V:H\rightarrow \text{End}_{\mathbbm{k}}(V)$ given by $\rho_V(h)(v)=h\cdot v$ for $h\in H$ and $v\in V$. We will make no distinction between the two notions. The character of $V$ is the map $\chi_V:H\rightarrow \mathbbm{k}$ given by $\chi_V(h)=\text{tr}(\rho_V(h))$ for $h\in H$.
The $\mathbbm{k}$-linear dual space $V^*$ is also an $H$-module via $(h\cdot f)(v):=f(S(h)\cdot v)$ for $h\in H$, $f\in V^*$ and $v\in V$.
In particular, the dual module $V^*$ has the character $\chi_{V^*}=\chi_V\circ S$. The category Rep$(H)$ of finite dimensional representations of $H$ is a semisimple tensor category, where the monoidal structure stems from the comultiplication $\Delta$.

Recall that the dual Hopf algebra $H^*$ has an $H$-bimodule structure given by $$(a\rightharpoonup f)(b)=f(ba),\ (f\leftharpoonup a)(b)=f(ab)\ \text{for}\ a,b\in H,\ f\in H^*.$$ Moreover, $(H^*,\leftharpoonup)$ and $(\rightharpoonup,H^*)$ are free $H$-modules generated by $\lambda$, i.e., $H^*=\lambda\leftharpoonup H$ and $H^*=H\rightharpoonup \lambda$ (see \cite[Corollary 2(b)]{Rad}). This provides an associative and non-degenerate bilinear form $H\times H\rightarrow \mathbbm{k}$ by $a\times b\mapsto\lambda(ab)$ for $a,b\in H$. Moreover, the pair $(H,\lambda)$ is a Frobenius algebra with the Frobenius homomorphism $\lambda$ satisfying the equality (see \cite[Eq.(1)]{Rad}): \begin{equation}gin{equation}\label{equ01}a=\lambda(a\Lambda_{(1)})S(\Lambda_{(2)})=\lambda(S(\Lambda_{(2)})a)\Lambda_{(1)}\ \text{for}\ a\in H.\end{equation} The pair $\Lambda_{(1)}\otimes S(\Lambda_{(2)})$ satisfying (\ref{equ01}) is called the dual basis of $H$ with respect to the Frobenius homomorphism $\lambda$.

Since the right integral $\lambda$ of $H^*$ satisfies  $\lambda(ab)=\lambda(S^2(b)a)$ for all $a,b\in H$ (see \cite[Theorem 3(a)]{Rad}), the Hopf algebra $H$ is a symmetric algebra with a symmetric bilinear form given by  $$H\times H\rightarrow \mathbbm{k},\ a\times b\mapsto\lambda(uab)=(\lambda\leftharpoonup u)(ab)=(u\rightharpoonup\lambda)(ab),$$ where $u$ is a unit of $H$ satisfying $S^2(h)=uhu^{-1}$ for all $h\in H$ and the Frobenius homomorphism $\lambda\leftharpoonup u=u\rightharpoonup\lambda$ holds because $\lambda(au)=\lambda(S^2(u)a)=\lambda(ua)$ for all $a\in H$. Using (\ref{equ01}) we may see that the pair $\Lambda_{(1)}\otimes u^{-1}S(\Lambda_{(2)})$ is a dual basis of $H$ with respect to $\lambda\leftharpoonup u\ (=u\rightharpoonup\lambda)$ (see also \cite[Lemma 1.4(2)]{Doi}). The symmetry of the Frobenius homomorphism $\lambda\leftharpoonup u\ (=u\rightharpoonup\lambda)$ means that \begin{equation}gin{equation}\label{equ5}\Lambda_{(1)}\otimes u^{-1}S(\Lambda_{(2)})= u^{-1}S(\Lambda_{(2)})\otimes \Lambda_{(1)}.\end{equation}

By Wedderburn's theorem, the semisimple Hopf algebra $H$ is isomorphic to a direct sum of full matrix algebras over $\mathbbm{k}$, namely,
$$H\cong\bigoplus_{i\in I}M_{d_i}(\mathbbm{k}).$$ Let $e_i$ be the idempotent of $H$ satisfying that $He_i\cong M_{d_i}(\mathbbm{k})$. Then $\{e_i\}_{i\in I}$ forms a complete set of central primitive idempotents of $H$.
Let $V_i$ be a simple left module (unique up to isomorphism) over the matrix algebra $M_{d_i}(\mathbbm{k})$. Then $\dim_{\mathbbm{k}}(V_i)=d_i$ and $\{V_i\}_{i\in I}$ forms a complete set of simple left $H$-modules up to isomorphism. The left regular representation $H$ has the decomposition $H\cong\bigoplus_{i\in I}V_i^{\oplus d_i}$ as $H$-modules, so the character $\chi_{H}$ of the left regular representation $H$ is equal to $\sum_{i\in I}d_i\chi_i$, where each $\chi_i$ is the character of $V_i$.

For any simple $H$-module $V_i$ and any $\varphi\in\text{End}_{\mathbbm{k}}(V_i)$, we use the dual basis $\Lambda_{(1)}\otimes u^{-1}S(\Lambda_{(2)})$ with respect to the Frobenius homomorphism $\lambda\leftharpoonup u$ to define the map $\mathcal{I}(\varphi)\in\text{End}_{\mathbbm{k}}(V_i)$ by $$\mathcal{I}(\varphi)(v)=\Lambda_{(1)}\varphi(u^{-1}S(\Lambda_{(2)})v)\ \text{for}\ v\in V_i.$$

Note that $\mathcal{I}(\varphi)$ lies in $\text{End}_{H}(V_i)\cong \mathbbm{k}$. There exists a unique element $c_{i}\in \mathbbm{k}$ such that
\begin{equation}gin{equation}\label{equ2}\mathcal{I}(\varphi)=c_i\text{tr}(\varphi)id_{V_i}\ \text{for\ all}\ \varphi\in\text{End}_{k}(V_i).\end{equation}
Such an element $c_i$, depending only on the isomorphism class of $V_i$, is called the Schur element associated to $V_i$ (see \cite[Theorem 7.2.1]{GP}). Since $H$ is semisimple, it follows from \cite[Theorem 7.2.6]{GP} that the Schur element $c_i\neq0$ in $\mathbbm{k}$
and the Frobenius homomorphism $\lambda\leftharpoonup u$  can be written explicitly as follows: \begin{equation}gin{equation}\label{equ4}\lambda\leftharpoonup u=u\rightharpoonup\lambda=\sum_{i\in I}\frac{1}{c_i}\chi_i.\end{equation}

\section{\bf A formula for the square of antipodes}
In this section, we will provide a formula for $S^2$ by virtue of a nonzero integral $\Lambda$ of $H$. Then we study some properties of the element  $\mathbf{u}:=S(\Lambda_{(2)})\Lambda_{(1)}$. Especially, we will give a sufficient and necessary condition for $S^2=id$ via the integral $\Lambda$.

Let $u$ be a unit of $H$ satisfying  $S^2(a)=uau^{-1}$ for all $a\in H$. We fix a left integral $\Lambda$ of $H$ and a right integral $\lambda$ of $H^*$ such that $\lambda(\Lambda)=1$. We denote $\{V_i\}_{i\in I}$ the set of all simple left $H$-modules up to isomorphism. For each $V_i$ we denote $c_i$ the Schur element of $V_i$ associated to the dual basis $\Lambda_{(1)}\otimes u^{-1}S(\Lambda_{(2)})$ of $H$ with respect to the Frobenius homomorphism $\lambda\leftharpoonup u$. We denote $\{e_i\}_{i\in I}$ the set of all central primitive idempotents of $H$.
We first establish a relationship between the elements $u$ and $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$.

\begin{equation}gin{prop}\label{th1} With the notions above, we have
$\mathbf{u}=u\sum_{i\in I}\dim_{\mathbbm{k}}(V_i)c_ie_i$, which is a unit of $H$.
\end{prop}
\proof Note that each central primitive idempotent $e_i$ acts as the identity on $V_i$ and annihilates $V_j$ for $j\neq i$. It follows that $\chi_j(e_i)=\dim_{\mathbbm{k}}(V_i)$ if $i=j$ and 0 otherwise. By (\ref{equ4}) we have
$$\chi_i(a)=\chi_i(ae_i)=\sum_{j\in I}\frac{1}{c_j}\chi_j(c_iae_i)=(u\rightharpoonup\lambda)(c_iae_i)=(uc_ie_i\rightharpoonup\lambda)(a).$$ Thus, $\chi_i=uc_ie_i\rightharpoonup\lambda$ and hence \begin{equation}gin{equation}\label{equ3}\chi_{H}=\sum_{i\in I}\dim_{\mathbbm{k}}(V_i)\chi_i=u\sum_{i\in I}\dim_{\mathbbm{k}}(V_i)c_ie_i\rightharpoonup\lambda.\end{equation}
For any map $\varphi\in\text{End}_{\mathbbm{k}}(H)$, the trace of $\varphi$ is $\text{tr}(\varphi)=\lambda(\varphi(S(\Lambda_{(2)}))\Lambda_{(1)})$ (see \cite[Theorem 2]{Rad}). Taking into account that $\varphi=L_a$, where $L_a$ is the left multiplication operator of $H$ by $a$, we have
$$\chi_H(a)=\text{tr}(L_a)=\lambda(aS(\Lambda_{(2)})\Lambda_{(1)})=(S(\Lambda_{(2)})\Lambda_{(1)}\rightharpoonup\lambda)(a).$$ This implies that $\chi_H=S(\Lambda_{(2)})\Lambda_{(1)}\rightharpoonup\lambda.$ Comparing it with (\ref{equ3}) and using the non-degeneracy of the Frobenius homomorphism $\lambda$, we have $$S(\Lambda_{(2)})\Lambda_{(1)}=u\sum_{i\in I}\dim_{\mathbbm{k}}(V_i)c_ie_i.$$ Since $p>\dim_{\mathbbm{k}}(H)^{1/2}$, it follows that $p^2>\dim_{\mathbbm{k}}(H)=\sum_{i\in I}\dim_{\mathbbm{k}}(V_i)^2\geq\dim_{\mathbbm{k}}(V_i)^2$. Hence $p>\dim_{\mathbbm{k}}(V_i)$ and $\dim_{\mathbbm{k}}(V_i)\neq0$ in $\mathbbm{k}$ for any $i\in I$.
Thus, the element $u$ is the same as $S(\Lambda_{(2)})\Lambda_{(1)}$ up to a central unit $\sum_{i\in I}\dim_{\mathbbm{k}}(V_i)c_ie_i$.
\qed

\begin{equation}gin{rem} Proposition \ref{th1} also holds if the field $\mathbbm{k}$ has characteristic 0. In this case,  $S^2=id$ (see \cite{LR1} or \cite{LR}) implying that  $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}=S(\Lambda_{(2)})S^2(\Lambda_{(1)})=S(S(\Lambda_{(1)})\Lambda_{(2)})=\varepsilon(\Lambda)$.\end{rem}

Proposition \ref{th1} gives a formula for $S^2$, namely,  $$S^2(a)=\mathbf{u}a\mathbf{u}^{-1}\ \text{for}\ a\in H,$$ where $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}.$
In the sequel, we will replace $u$ with $\mathbf{u}$. In this case, the equality (\ref{equ5}) turns out to be \begin{equation}gin{equation}\label{equ55}\Lambda_{(1)}\otimes \mathbf{u}^{-1}S(\Lambda_{(2)})=\mathbf{u}^{-1}S(\Lambda_{(2)})\otimes \Lambda_{(1)},\end{equation}
which is the dual basis of $H$ with respect to the Frobenius homomorphism $\lambda\leftharpoonup \mathbf{u}$.
The Schur element associated to the simple $H$-module $V_i$ under the new dual basis $\Lambda_{(1)}\otimes \mathbf{u}^{-1}S(\Lambda_{(2)})$ with respect to the Frobenius homomorphism $\lambda\leftharpoonup \mathbf{u}$ is $\frac{1}{\dim_{\mathbbm{k}}(V_i)}$.
Therefore, the equality (\ref{equ4}) turns out to be
\begin{equation}gin{equation}\label{equ0077}\lambda\leftharpoonup \mathbf{u}=\mathbf{u}\rightharpoonup\lambda=\sum_{i\in I}\dim_{\mathbbm{k}}(V_i)\chi_i=\chi_H.\end{equation}
By applying \cite[Theorem 1.5]{Doi} and (\ref{equ55}), we obtain the expression of
each central primitive idempotent $e_i$ of $H$ as follows:
\begin{equation}gin{equation}\label{equ8-112}e_i=\dim_{\mathbbm{k}}(V_i)\chi_i(\Lambda_{(1)})\mathbf{u}^{-1}S(\Lambda_{(2)})=\dim_{\mathbbm{k}}(V_i) \chi_i(\mathbf{u}^{-1}S(\Lambda_{(2)}))\Lambda_{(1)}.\end{equation}

Let $g\in G(H)$ and $\alphapha\in\text{Alg}(H,k)$ be the modular elements of $H$ and $H^*$ respectively. Recall that the Radford's formula of $S^4$ has the form (see \cite[Proposition 6]{Rad1}): $$S^4(a)=\alphapha^{-1}\rightharpoonup(gag^{-1})\leftharpoonup\alphapha.$$ Since $H$ is unimodular, i.e., $\alphapha=\varepsilon$, the Radford's formula of $S^4$ now becomes $$S^4(a)=gag^{-1}.$$
The distinguished group-like element $g$ and the integral $\Lambda$ of $H$ satisfy the following useful equality (see \cite[Theorem 3(d)]{Rad}):
\begin{equation}gin{equation}\label{equ0}\Lambda_{(2)}\otimes\Lambda_{(1)}=\Lambda_{(1)}\otimes S^2(\Lambda_{(2)})g.\end{equation}
After these preparations, we give some properties of the element $\mathbf{u}$ as follows:

\begin{equation}gin{prop}\label{prop3.20}
The element $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$ satisfies the the following properties:
\begin{equation}gin{enumerate}
  \item $\mathbf{u}=\chi_{H}(\Lambda_{(1)})S(\Lambda_{(2)})$.
  \item $\Lambda_{(1)}\mathbf{u}^{-1} S(\Lambda_{(2)})=1.$
  \item $\lambda(e_i)=\dim_{\mathbbm{k}}(V_i)\chi_i(\mathbf{u}^{-1})$.
  \item $\mathbf{u}S(\mathbf{u})=S(\mathbf{u})\mathbf{u}=\varepsilon(\Lambda)\sum_{i\in I}\frac{\dim_{\mathbbm{k}}(V_i)^2}{\lambda(e_i)}e_i$.
  \item $S(\mathbf{u}^{-1})\mathbf{u}=\mathbf{u}S(\mathbf{u}^{-1})$, which is the distinguished group-like element $g$ of $H$.
\end{enumerate}
\end{prop}
\proof (1) It follows from (\ref{equ8-112}) that $e_i\mathbf{u}=\dim_{\mathbbm{k}}(V_i)\chi_i(\Lambda_{(1)})S(\Lambda_{(2)})$. Thus, $$\mathbf{u}=\sum_{i\in I}e_i\mathbf{u}=\sum_{i\in I}\dim_{\mathbbm{k}}(V_i)\chi_i(\Lambda_{(1)})S(\Lambda_{(2)})=\chi_{H}(\Lambda_{(1)})S(\Lambda_{(2)}).$$

(2) Since $\Lambda_{(1)}\otimes \mathbf{u}^{-1}S(\Lambda_{(2)})=\mathbf{u}^{-1}S(\Lambda_{(2)})\otimes \Lambda_{(1)}$ by (\ref{equ55}), we obtain the desired result by multiplying the tensor factors together.

(3) Since $e_i=\dim_{\mathbbm{k}}(V_i)\chi_i(\Lambda_{(1)})\mathbf{u}^{-1}S(\Lambda_{(2)})$, it follows that $$e_i=\mathbf{u}e_i\mathbf{u}^{-1}=\dim_{\mathbbm{k}}(V_i)\chi_i(\Lambda_{(1)})S(\Lambda_{(2)})\mathbf{u}^{-1}.$$
Hence $$\lambda(e_i)=\dim_{\mathbbm{k}}(V_i)\chi_i(\Lambda_{(1)})\lambda(S(\Lambda_{(2)})\mathbf{u}^{-1})=\dim_{\mathbbm{k}}(V_i)\chi_i(\mathbf{u}^{-1}),$$
where the last  equality follows from (\ref{equ01}).

(4) For any $a\in H$, we have $S^3(a)=S(S^2(a))=S(\mathbf{u} a\mathbf{u}^{-1})=S(\mathbf{u}^{-1})S(a)S(\mathbf{u})$, we also have $S^3(a)=S^2(S(a))=\mathbf{u}S(a)\mathbf{u}^{-1}$. It follows that $S(\mathbf{u})\mathbf{u}$ is a central unit of $H$. The equality $\mathbf{u}S(\mathbf{u})=S(\mathbf{u})\mathbf{u}$ holds because $S(\mathbf{u})=S(S^2(\mathbf{u}))=S^2(S(\mathbf{u}))=\mathbf{u}S(\mathbf{u})\mathbf{u}^{-1}$.
For the central unit $\mathbf{u}S(\mathbf{u})$, we suppose that $\mathbf{u}S(\mathbf{u})=\sum_{i\in I} k_ie_i$, where each scalar $k_i\neq0$ in $\mathbbm{k}$. Then $e_i\mathbf{u}^{-1}=\frac{1}{k_i}e_iS(\mathbf{u})$. We have
\begin{equation}gin{align*}\lambda(e_i)&=(\mathbf{u}^{-1}\rightharpoonup\chi_H)(e_i)=\chi_H(e_i\mathbf{u}^{-1})=\frac{1}{k_i}\chi_H(e_iS(\mathbf{u}))\\
&=\frac{\dim_{\mathbbm{k}}(V_i)}{k_i}\chi_i(e_iS(\mathbf{u}))=\frac{\dim_{\mathbbm{k}}(V_i)}{k_i}\chi_i(S(\mathbf{u}))\\
&=\frac{\dim_{\mathbbm{k}}(V_i)}{k_i}(\chi_i\circ S)(\mathbf{u})=\frac{\dim_{\mathbbm{k}}(V_i)}{k_i}(\chi_i\circ S)( S(\Lambda_{(2)})\Lambda_{(1)})\\
&=\frac{\dim_{\mathbbm{k}}(V_i)}{k_i}(\chi_i\circ S)(\Lambda_{(1)}S(\Lambda_{(2)}))=\frac{\dim_{\mathbbm{k}}(V_i)^2\varepsilon(\Lambda)}{k_i}\neq0.
\end{align*}
It follows that $k_i=\frac{\dim_{\mathbbm{k}}(V_i)^2\varepsilon(\Lambda)}{\lambda(e_i)}$ and $\mathbf{u}S(\mathbf{u})=\sum_{i\in I} k_ie_i=\varepsilon(\Lambda)\sum_{i\in I} \frac{\dim_{\mathbbm{k}}(V_i)^2}{\lambda(e_i)}e_i$.

(5) Note that $\Lambda_{(2)}\otimes\Lambda_{(1)}=\Lambda_{(1)}\otimes S^2(\Lambda_{(2)})g$ by (\ref{equ0}). Applying $S\otimes id$ to both sides of this equality and multiplying the tensor factors together, we have $\mathbf{u}=S(\mathbf{u})g$ or $g=S(\mathbf{u}^{-1})\mathbf{u}$.
\qed

As a consequence, we obtain the following result:

\begin{equation}gin{cor}\label{cor4}For any central primitive idempotent $e_i$ of $H$,
we have $\lambda(e_i)=\lambda(S(e_i))$.
\end{cor}
\proof  We denote $S(e_i)=e_{i^*}$ for some $i^*\in I$, then  $V_i^*\cong V_{i^*}$, or equivalently, $\chi_i\circ S=\chi_{i^*}$ (see \cite[Lemma 1.8]{Doi}).
By Proposition \ref{prop3.20} (3) we have $$\lambda(S(e_i))=\lambda(e_{i^*})=\dim_{\mathbbm{k}}(V_{i^*})\chi_{i^*}(\mathbf{u}^{-1})=\dim_{\mathbbm{k}}(V_i)\chi_i(S(\mathbf{u}^{-1})).$$
Since $\mathbf{u}S(\mathbf{u})=\varepsilon(\Lambda)\sum_{i\in I} \frac{\dim_{\mathbbm{k}}(V_i)^2}{\lambda(e_i)}e_i$, it follows that $S(\mathbf{u}^{-1})=\mathbf{u}\frac{1}{\varepsilon(\Lambda)}\sum_{i\in I} \frac{\lambda(e_i)}{\dim_{\mathbbm{k}}(V_i)^2}e_i$. Thus,
\begin{equation}gin{align*}\lambda(S(e_i))&=\dim_{\mathbbm{k}}(V_i)\chi_i(S(\mathbf{u}^{-1}))=\frac{\lambda(e_i)}{\varepsilon(\Lambda)\dim_{\mathbbm{k}}(V_i)}\chi_i(\mathbf{u})\\
&=\frac{\lambda(e_i)}{\varepsilon(\Lambda)\dim_{\mathbbm{k}}(V_i)}\chi_i(\Lambda_{(1)}S(\Lambda_{(2)}))=\lambda(e_i).
\end{align*}
We complete the proof. \qed

If the field $\mathbbm{k}$ has characteristic 0, then the antipode $S$ of $H$ satisfies $S^2=id$ (see \cite{LR1} or \cite{LR}). This further implies that the integral $\Lambda$ of $H$ is cocommutative (see \cite[Proposition 2(b)]{LR1}). The following result shows that $\Lambda$ being cocommutative is equivalent to $S^2=id$ when the characteristic of the field $\mathbbm{k}$ is larger then $\dim_{\mathbbm{k}}(H)^{1/2}$.

\begin{equation}gin{prop}\label{thm4}Let $H$ be a finite dimensional semisimple Hopf algebra over the field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. The following statements are equivalent:
\begin{equation}gin{enumerate}
  \item The nonzero integral $\Lambda$ of $H$ is cocommutative.
  \item The nonzero integral $\lambda$ of $H^*$ is cocommutative.
  \item $S^2=id$.
\end{enumerate}
\end{prop}
\proof It can be seen from  \cite[Corollary 5]{Rad} that Part (2) and Part (3) are equivalent. We next show that Part (1) and Part (3) are equivalent. If $\Lambda$ is cocommutative, then $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}= S(\Lambda_{(1)})\Lambda_{(2)}=\varepsilon(\Lambda)$. It follows from $S^2(a)=\mathbf{u}a\mathbf{u}^{-1}$ that $S^2=id$. Conversely, if $S^2=id$, then $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}=S(\Lambda_{(2)})S^2(\Lambda_{(1)})=S( S(\Lambda_{(1)})\Lambda_{(2)})=\varepsilon(\Lambda)$. By Proposition \ref{prop3.20}, we have $g=S(\mathbf{u}^{-1})\mathbf{u}=1$. Since $\Lambda_{(2)}\otimes\Lambda_{(1)}=\Lambda_{(1)}\otimes S^2(\Lambda_{(2)})g$ by (\ref{equ0}), it follows that $\Lambda_{(2)}\otimes\Lambda_{(1)}=\Lambda_{(1)}\otimes \Lambda_{(2)}$. We complete the proof.
\qed

\section{\bf Higher FS indicators}
If the field $\mathbbm{k}$ has characteristic 0, the $n$-th FS indicators of finite dimensional representations of semisimple Hopf algebras have been studied in \cite{KSZ}. In this section, we will generalize these indicators from characteristic 0 to the case of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$ and describe them via a nonzero integral $\Lambda$ of $H$. We begin with the following preparations.

Let $H$ be a finite dimensional semisimple Hopf algebra over the field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$ with a nonzero integral $\Lambda$ and $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$.
Applying $\Delta_{n-1}\otimes id$ to both sides of the equality: $\Lambda_{(2)}\otimes\Lambda_{(1)}=\Lambda_{(1)}\otimes S^2(\Lambda_{(2)})g$ (see (\ref{equ0})), we have
\begin{equation}gin{equation*}\Lambda_{(2)}\otimes\cdots\otimes\Lambda_{(n)}\otimes\Lambda_{(1)}=\Lambda_{(1)}\otimes\cdots \otimes\Lambda_{(n-1)} \otimes S^2(\Lambda_{(n)})g.\end{equation*}
Since $g=\mathbf{u}S(\mathbf{u}^{-1})$ and $S^2(\Lambda_{(n)})=\mathbf{u}\Lambda_{(n)}\mathbf{u}^{-1}$, the above equality induces the following equality:
\begin{equation}gin{equation}\label{equ00}\Lambda_{(2)}\otimes\cdots\otimes\Lambda_{(n)}\otimes \mathbf{u}^{-1}\Lambda_{(1)}=\Lambda_{(1)}\otimes\cdots \otimes\Lambda_{(n-1)} \otimes \Lambda_{(n)}S(\mathbf{u}^{-1}).\end{equation}

Note that the category Rep$(H)$ of finite dimensional representations of $H$ is a semisimple tensor category. Let $j_{\mathbf{u}}:id\rightarrow(-)^{**}$ be a natural isomorphism between the identity functor and
the functor of taking the second dual. It is completely determined by a collection of $H$-module isomorphisms
$$j_{\mathbf{u},V}:V\rightarrow V^{**},\ \ j_{\mathbf{u},V}(v)(f)=f(\mathbf{u}v)\ \text{for}\ v\in V, f\in V^*.$$
The inverse of $j_{\mathbf{u},V}$ is $$j_{\mathbf{u},V}^{-1}:V^{**}\rightarrow V,\ \ \alphapha\mapsto j_{\mathbf{u},V}^{-1}(\alphapha),$$ where $j_{\mathbf{u},V}^{-1}(\alphapha)\in V$ satisfies the equality  $f(j_{\mathbf{u},V}^{-1}(\alphapha))=\alphapha(S^{-1}(\mathbf{u}^{-1})f)$ for $f\in V^*$. Since $S^2(h)=\mathbf{u}h\mathbf{u}^{-1}$ and $\mathbf{u}$ is not known to be a group-like element, the natural isomorphism $j_{\mathbf{u}}$ is not necessarily a tensor isomorphism.
Although the representation category Rep$(H)$ with respect to the structure $j_{\mathbf{u}}$ is not necessarily pivotal, we may still define higher FS indicators for any finite dimensional representation of $H$  using the structure $j_{\mathbf{u}}$ of Rep$(H)$.

We denote $V^{\otimes n}$ the $n$-th tensor power of $V$ where $V^{\otimes0}$ is the trivial $H$-module $\mathbbm{k}$.
For any natural number $n\geq1$,
we define the following $\mathbbm{k}$-linear map $$E_V^{n}:\text{Hom}_H(\mathbbm{k},V^{\otimes n})\rightarrow \text{Hom}_H(\mathbbm{k},V^{\otimes n}),\ \ f\mapsto E_V^{n}(f),$$
where $E_V^{n}(f)$ is an $H$-module morphism from $\mathbbm{k}$ to $V^{\otimes n}$ given by
$$E_V^{n}(f):\mathbbm{k}\xrightarrow{\text{coev}_{V^*}}V^*\otimes V^{**}=V^*\otimes \mathbbm{k}\otimes V^{**}\xrightarrow{id\otimes f\otimes id}
V^*\otimes V^{\otimes n}\otimes V^{**}$$
$$\xrightarrow{\text{ev}_V\otimes id}V^{\otimes (n-1)}\otimes V^{**}\xrightarrow{id\otimes j_{\mathbf{u},V}^{-1}}V^{\otimes n}.$$
Here the maps $\text{coev}_{V^*}$ and $\text{ev}_V$ are the usual coevaluation morphism of $V^*$ and evaluation morphism of $V$ respectively.
If we set $f(1)=\sum v_1\otimes\cdots\otimes v_n\in V^{\otimes n}$, the above definition of $E_V^{n}(f)$ shows that
\begin{equation}gin{equation}\label{equ09}E_V^{n}(f)(1)=\sum v_2\otimes\cdots\otimes v_n\otimes \mathbf{u}^{-1}v_1.\end{equation}

Similar to \cite{NS2}, we give the definition of the $n$-th FS indicator of $V$ to be the trace of the linear operator $E_V^{n}$ as follows:

\begin{equation}gin{defi}Let $H$ be a finite dimensional semisimple Hopf algebra over the field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$.
For any finite dimensional representation $V$ of $H$, the $n$-th FS indicator of $V$ is defined by
$$\nu_n(V)=\text{tr}(E_V^{n})\ \text{for}\ n\geq1.$$
\end{defi}

Similar to the characteristic 0 case, the $n$-th FS indicator of $V$ defined above can also be described by a nonzero integral $\Lambda$ of $H$:

\begin{equation}gin{thm}\label{prop3}Let $\Lambda$ be a nonzero integral of $H$  and $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$. Suppose $\chi_V$ is the character of a finite dimensional representation $V$ of $H$.
We have $$\nu_n(V)=\chi_V(\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)})\ \text{for}\ n\geq1.$$
\end{thm}
\proof We first show that the equality $\nu_n(V)=\chi_V(\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)})$ holds for an idempotent integral $\Lambda$.
Suppose that $\alphapha$ is the following $\mathbbm{k}$-linear map
$$\alphapha:V^{\otimes n}\rightarrow V^{\otimes n},\ \ v_1\otimes v_2\otimes\cdots \otimes v_n\mapsto v_2\otimes\cdots \otimes v_n\otimes v_1$$ and $\delta=\alphapha\circ(\mathbf{u}^{-1}\Lambda_{(1)}\otimes \Lambda_{(2)}\otimes\cdots\otimes\Lambda_{(n)}).$ We have
\begin{equation}gin{align}\label{equ29}\nonumber\delta(v_1\otimes v_2\otimes\cdots \otimes v_n)&=\alphapha(\mathbf{u}^{-1}\Lambda_{(1)}v_1\otimes \Lambda_{(2)}v_2\otimes \cdots\otimes\Lambda_{(n)}v_n)\\
\nonumber&=\Lambda_{(2)}v_2\otimes\cdots\otimes \Lambda_{(n)}v_n\otimes \mathbf{u}^{-1}\Lambda_{(1)}v_1\\
&=\Lambda_{(1)}v_2\otimes\cdots\otimes \Lambda_{(n-1)}v_n\otimes \Lambda_{(n)}S(\mathbf{u}^{-1})v_1\ \text{by}\ (\ref{equ00})\\
\nonumber&=\Lambda\cdot(v_2\otimes\cdots\otimes v_n\otimes S(\mathbf{u}^{-1})v_1).
\end{align}
This shows that $\delta(V^{\otimes n})\subseteq \Lambda\cdot V^{\otimes n}=(V^{\otimes n})^H.$
Note that the map $$\Phi:\textrm{Hom}_H(\mathbbm{k},V^{\otimes n})\rightarrow(V^{\otimes n})^H,\ \ f\mapsto f(1)$$ is an $H$-module isomorphism. We claim that the following diagram is commutative:
\begin{equation}gin{linenomath}
\begin{equation}gin{equation*}
\xymatrix{
  \textrm{Hom}_H(\mathbbm{k},V^{\otimes n}) \ar[d]_{\Phi} \ar[r]^{E_V^{n}}& \textrm{Hom}_H(\mathbbm{k},V^{\otimes n})\ar[d]^{\Phi}& \\
 (V^{\otimes n})^H \ar[r]^{\delta}&(V^{\otimes n})^H.&}
\end{equation*}
\end{linenomath}
Indeed, for any $f\in\text{Hom}_H(\mathbbm{k},V^{\otimes n})$, we suppose that $f(1)=\sum v_1\otimes\cdots\otimes v_n\in V^{\otimes n}$. It follows from $f(1)=f(\Lambda\cdot1)=\Lambda\cdot f(1)$ that
\begin{equation}gin{equation}\label{equ19}\sum v_1\otimes\cdots\otimes v_n=\sum\Lambda_{(1)}v_1\otimes\cdots\otimes\Lambda_{(n)}v_n.\end{equation}
On the one hand, we have
\begin{equation}gin{align*}
(\delta\circ\Phi)(f)&=\delta(f(1))=\delta(\sum v_1\otimes\cdots\otimes v_n)\\
&=\Lambda\cdot(\sum v_2\otimes\cdots\otimes v_n\otimes S(\mathbf{u}^{-1})v_1)\ \ \text{by}\ (\ref{equ29})
\end{align*}
On the other hand, we have
\begin{equation}gin{align*}
(\Phi\circ E_V^n)(f)&=E_V^n(f)(1)=\sum v_2\otimes\cdots\otimes v_n\otimes \mathbf{u}^{-1}v_1\ \ \ \text{by}\ (\ref{equ09})\\
&=\sum \Lambda_{(2)}v_2\otimes\cdots\otimes \Lambda_{(n)}v_n\otimes \mathbf{u}^{-1}\Lambda_{(1)}v_1\ \ \ \text{by}\ (\ref{equ19})\\
&=\sum \Lambda_{(1)}v_2\otimes\cdots\otimes \Lambda_{(n-1)}v_n\otimes\Lambda_{(n)}S(\mathbf{u}^{-1})v_1\ \ \ \text{by}\ (\ref{equ00})\\
&=\Lambda\cdot(\sum v_2\otimes\cdots\otimes v_n\otimes S(\mathbf{u}^{-1})v_1).
\end{align*}
We obtain that $\delta\circ\Phi=\Phi\circ E_V^n$, or equivalently, $E_V^n=\Phi^{-1}\circ\delta\circ\Phi$. It follows that
\begin{equation}gin{align*}\nu_n(V)&=\text{tr}(E_V^{n})=\text{tr}_{V^{\otimes n}}(\delta)\\
&=\text{tr}_{V^{\otimes n}}(\alphapha\circ(\mathbf{u}^{-1}\Lambda_{(1)}\otimes \Lambda_{(2)}\otimes\cdots\otimes\Lambda_{(n)}))\\
&=\text{tr}_V(\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)})\\
&=\chi_V(\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)}),
\end{align*}
where the equality $\text{tr}_{V^{\otimes n}}(\alphapha\circ(\mathbf{u}^{-1}\Lambda_{(1)}\otimes \Lambda_{(2)}\otimes\cdots\otimes\Lambda_{(n)}))
=\text{tr}_V(\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)})$ follows from \cite[Lemma 2.3]{KSZ}. We have shown that  $\nu_n(V)=\chi_V(\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)})$ where $\Lambda$ is idempotent. Since $\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)}$ does not depend on the choice of the nonzero integral $\Lambda$, the equality $\nu_n(V)=\chi_V(\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)})$ holds for any nonzero integral $\Lambda$ of $H$.
\qed

\begin{equation}gin{rem}
If the field $\mathbbm{k}$ has characteristic 0 and $\Lambda$ is idempotent, then $\mathbf{u}=\varepsilon(\Lambda)=1$. In this case, the $n$-th FS indicator of $V$ is $\chi_V(\Lambda_{(1)}\cdots\Lambda_{(n)})$, which is  the one defined in \cite[Definition 2.3]{KSZ}.
\end{rem}

In the rest of this section, we will extend the $n$-th FS indicator $\nu_n(V)$ of $V$ from $n\geq1$ to the case $n\in\mathbb{Z}$.
Recall that the $n$-th Sweedler power map $P_n:H\rightarrow H$ is defined by
$$P_n(a)=\left\{
           \begin{equation}gin{array}{ll}
             a_{(1)}\cdots a_{(n)}, &  n\geq1; \\
             \varepsilon(a), & n=0; \\
             S(a_{(1)})\cdots S(a_{(-n)}), &  n\leq-1.
           \end{array}
         \right.
$$ From the $n$-th Sweedler power map $P_n$ of $H$, we may see that $$\nu_n(V)=\chi_V(\mathbf{u}^{-1}P_n(\Lambda))\ \text{for}\ n\geq1.$$
However, this expression is well-defined for any integer $n$. Thus, we may extend this formula from $n\geq1$ to any integer $n$ stated as follows:
\begin{equation}gin{defi}Let $H$ be a finite dimensional semisimple Hopf algebra over the field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$.
For any finite dimensional representation $V$ of $H$ and any $n\in\mathbb{Z}$, the $n$-th FS indicator of $V$ is defined by
$$\nu_n(V)=\chi_V(\mathbf{u}^{-1}P_n(\Lambda)),$$
where $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$.
\end{defi}

\begin{equation}gin{rem}\label{rem1}
\begin{equation}gin{enumerate}
\item Note that $S(\Lambda)=\Lambda$. The $n$-th FS indicator of $V$  can be written as
$$\nu_n(V)=\left\{
             \begin{equation}gin{array}{ll}
               \chi_V(\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)}), & n\geq1; \\
               \chi_V(\mathbf{u}^{-1}\varepsilon(\Lambda)), & n=0; \\
               \chi_V(\mathbf{u}^{-1}\Lambda_{(-n)}\cdots\Lambda_{(1)}), & n\leq-1.
             \end{array}
           \right.
$$
\item By Proposition \ref{prop3.20} (4), we have \begin{equation}gin{equation*}\mathbf{u}^{-1}S(\mathbf{u}^{-1})=\sum_{i\in I}\frac{\lambda(e_i)}{\varepsilon(\Lambda)\dim_{\mathbbm{k}}(V_i)^2}e_i\in Z(H).\end{equation*} It follows that  \begin{equation}gin{align*}\nu_0(V)&=\varepsilon(\Lambda)\chi_V(\mathbf{u}^{-1})=\varepsilon(\Lambda)\chi_V(\mathbf{u}^{-1}S(\mathbf{u}^{-1})S(\mathbf{u}))\\
&=\sum_{i\in I}\frac{\lambda(e_i)}{\dim_{\mathbbm{k}}(V_i)^2}\chi_V(e_iS(\mathbf{u}))=\sum_{i\in I}\frac{\lambda(e_i)}{\dim_{\mathbbm{k}}(V_i)^2}\chi_V(e_iS(\Lambda_{(1)})S^2(\Lambda_{(2)}))\\
&=\sum_{i\in I}\frac{\lambda(e_i)}{\dim_{\mathbbm{k}}(V_i)^2}\chi_V(e_iS^2(\Lambda_{(2)})S(\Lambda_{(1)}))=\varepsilon(\Lambda)\sum_{i\in I}\frac{\lambda(e_i)}{\dim_{\mathbbm{k}}(V_i)^2}\chi_V(e_i).
\end{align*}
  \item $\nu_{-1}(V)=\nu_{1}(V)=\chi_V(\mathbf{u}^{-1}\Lambda)=\chi_V(\frac{\Lambda}{\varepsilon(\Lambda)})$.
  \item By \cite[Proposition 3.1]{WLL}, $\Lambda_{(1)}\Lambda_{(2)}$ and $\Lambda_{(2)}\Lambda_{(1)}$ are both central elements of $H$, they are determined by the values that the characters $\chi_i$ for all $i\in I$ take on them. It follows from $\chi_i(\Lambda_{(1)}\Lambda_{(2)})=\chi_i(\Lambda_{(2)}\Lambda_{(1)})$ that $\Lambda_{(1)}\Lambda_{(2)}=\Lambda_{(2)}\Lambda_{(1)}$. Therefore, $\nu_{-2}(V)=\nu_{2}(V)$.
\end{enumerate}
\end{rem}

The higher FS indicators of any simple module $V_i$ can be described as follows:
\begin{equation}gin{prop}For any $n\in\mathbb{Z}$ and any simple module $V_i$ with the character $\chi_i$, we have
$$\nu_n(V_i)=\frac{\chi_i(P_n(\Lambda))\lambda(e_i)}{\dim_{\mathbbm{k}}(V_i)^2}.
$$
\end{prop}
\proof
Since $P_n(\Lambda)\in Z(H)$ for any $n\in\mathbb{Z}$ (see \cite[Proposition 3.1]{WLL}), it follows that $P_n(\Lambda)=\sum_{i\in I}\frac{\chi_i(P_n(\Lambda))}{\dim_{\mathbbm{k}}(V_i)}e_i$. The $n$-th FS indicator of  $V_i$ is $$\nu_n(V_i)=\chi_i(\mathbf{u}^{-1}P_n(\Lambda))
=\frac{\chi_i(P_n(\Lambda))}{\dim_{\mathbbm{k}}(V_i)}\chi_i(\mathbf{u}^{-1})
=\frac{\chi_i(P_n(\Lambda))\lambda(e_i)}{\dim_{\mathbbm{k}}(V_i)^2},
$$
where the last equality follows from Proposition \ref{prop3.20} (3).
\qed

For any semisimple Hopf algebra over a field $\mathbbm{k}$ of characteristic 0, the finite dimensional representation $V$ and its dual $V^*$ have the same $n$-th FS indicators for all $n\geq1$ (see \cite[Section 2.3]{KSZ}). The following result shows that this result also holds for the $n$-th FS indicators defined for the Hopf algebra $H$ over the field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$.

\begin{equation}gin{prop}\label{prop3.21} Let $H$ be a finite dimensional semisimple Hopf algebra over the field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. Let $V$ be a finite dimensional representation of $H$ with the dual $V^*$. We have $\nu_n(V)=\nu_n(V^*)$ for all $n\in\mathbb{Z}$.
\end{prop}
\proof
Since $S(\Lambda)=\Lambda$, we have $S(P_n(\Lambda))=P_n(\Lambda)$ for any $n\in\mathbb{Z}$.
For the case $n\geq1$, the $n$-th FS indicator of $V^*$ is
\begin{equation}gin{align*}\nu_n(V^*)&=(\chi_{V^*})(\mathbf{u}^{-1}P_n(\Lambda))=(\chi_V\circ S)(\mathbf{u}^{-1}P_n(\Lambda))\\
&=\chi_V(\Lambda_{(1)}\cdots \Lambda_{(n)}S(\mathbf{u}^{-1}))=\chi_V(\Lambda_{(2)}\cdots\Lambda_{(n)}\mathbf{u}^{-1}\Lambda_{(1)})\ \ \text{by}\ (\ref{equ00})\\
&=\chi_V(\mathbf{u}^{-1}\Lambda_{(1)}\Lambda_{(2)}\cdots\Lambda_{(n)})=\nu_n(V).
\end{align*}

For the case $n\leq-1$,
the $n$-th FS indicator of $V^*$ is
\begin{equation}gin{align*}\nu_{n}(V^*)&=(\chi_{V^*})(\mathbf{u}^{-1}P_n(\Lambda))=(\chi_V\circ S)(\mathbf{u}^{-1}P_n(\Lambda))\\
&=\chi_V(\Lambda_{(-n)}\cdots \Lambda_{(1)}S(\mathbf{u}^{-1}))=\chi_V(S(\mathbf{u}^{-1})\Lambda_{(-n)}\cdots \Lambda_{(1)})\\
&=\chi_V(S(\mathbf{u}^{-1})\mathbf{u}^{-1}\Lambda_{(1)}S(\mathbf{u})\Lambda_{(-n)}\cdots\Lambda_{(2)})\ \ \text{by}\ (\ref{equ00})\\
&=\chi_V(\Lambda_{(1)}\mathbf{u}^{-1}\Lambda_{(-n)}\cdots\Lambda_{(2)})=\chi_V(\mathbf{u}^{-1}\Lambda_{(-n)}\cdots\Lambda_{(1)})\\
&=\nu_{n}(V).
\end{align*}

For the case $n=0$,
we denote $S(e_i)=e_{i^*}$ for any $i\in I$, then $*$ is a permutation of $I$,  $V_{i^*}\cong V_i^*$ and $\lambda(e_{i^*})=\lambda(e_i)$ by Corollary \ref{cor4}. We have
\begin{equation}gin{align*}\nu_0(V^*)&=\varepsilon(\Lambda)\sum_{i\in I}\frac{\lambda(e_i)}{\dim_{\mathbbm{k}}(V_i)^2}\chi_V(S(e_i))\ \ \text{by\ Remark}\ \ref{rem1} (2)\\
&=\varepsilon(\Lambda)\sum_{i\in I}\frac{\lambda(e_{i^*})}{\dim_{\mathbbm{k}}(V_{i^*})^2}\chi_V(e_{i^*})=\varepsilon(\Lambda)\sum_{i\in I}\frac{\lambda(e_{i})}{\dim_{\mathbbm{k}}(V_{i})^2}\chi_V(e_{i})\\
&=\nu_0(V).
\end{align*}
We complete the proof.
\qed

Kashina-Sommerh$\ddot{a}$user-Zhu has shown in \cite[Proposition 2.5]{KSZ} that the $n$-th FS indictor of the regular representation of a semisimple
Hopf algebra over a field of characteristic 0 can be described as $\text{tr}(S\circ P_{n-1})$ for $n\geq1.$
The following result shows that this formula also holds for the $n$-th FS indicators defined for the Hopf algebra $H$ over the field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$.

\begin{equation}gin{prop}Let $H$ be a finite dimensional semisimple Hopf algebra over the field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. For any $n\in\mathbb{Z}$, the $n$-th FS indictor of the regular representation of $H$ can be written as
 $\nu_n(H)=\text{tr}(S\circ P_{n-1}),$ where $P_{n-1}$ is the $(n-1)$-th Sweedler power map of $H$.
\end{prop}
\proof We choose a left integral $\Lambda$ of $H$ and a right integral $\lambda$ of $H^*$ such that $\lambda(\Lambda)=1$. For any $n\in\mathbb{Z}$, by Radford's trace formula \cite[Theorem 2]{Rad}, we have
\begin{equation}gin{align*}
\text{tr}(S\circ P_{n-1})&=\text{tr}(P_{n-1}\circ S)=\lambda(S(\Lambda_{(2)})(P_{n-1}\circ S)(\Lambda_{(1)}))\\
&=\lambda(S(\Lambda_{(2)})P_{n-1}(S(\Lambda_{(1)})))=\lambda(\Lambda_{(1)}P_{n-1}(\Lambda_{(2)}))\\
&=\lambda(P_n(\Lambda))=\chi_H(\mathbf{u}^{-1}P_n(\Lambda))\ \ \text{by}\ (\ref{equ0077})\\
&=\nu_{n}(H).
\end{align*}
We complete the proof. \qed

\section{\bf Gauge invariants}
In this section, we will show that the $n$-th FS indicator $\nu_n(V)$ defined in Section 4 is a gauge invariant of the tensor category Rep$(H)$ for any $n\in\mathbb{Z}$ and any finite dimensional representation $V$ of the semisimple Hopf algebra $H$ .

Recall from \cite{AEGN} that a  (normalized) twist for semisimple Hopf algebra $H$ is an invertible element $J\in H\otimes H$ that satisfies $(\varepsilon\otimes id)(J)=(id\otimes\varepsilon)(J)=1$ and
$$(\Delta\otimes id)(J)(J\otimes1)=(id\otimes\Delta)(J)(1\otimes J).$$
We write $J=J^{(1)}\otimes J^{(2)}$ and $J^{-1}=J^{-(1)}\otimes J^{-(2)}$, where the summation is understood. 

Given a twist $J$ for $H$ one can define a new Hopf algebra $H^{J}$ with the same algebra structure and counit as $H$, for which the comultiplication $\Delta^J$ and antipode $S^J$ are given respectively by
$$\Delta^J(a)=J^{-1}\Delta(a)J,$$
$$S^J(a)=Q^{-1}_JS(a)Q_J,\ \ \text{for}\ a\in H,$$
where $Q_J=S(J^{(1)})J^{(2)}$, which is an invertible element of $H$ with the inverse $Q_J^{-1}=J^{-(1)}S(J^{-(2)})$.
With the notions above, we have the following result:

\begin{equation}gin{prop}\label{thm5}Let $H$ be a finite dimensional semisimple Hopf algebra over the field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$ and $V$ a finite dimensional representation of $H$.
The $n$-th FS indicator $\nu_n(V)$ of $V$ is invariant under twisting for any $n\in\mathbb{Z}$.
\end{prop}
\proof
Let $\Lambda$ be a nonzero integral of $H$ and $J$ a normalized twist for $H$. It follows from \cite[Theorem 3.4]{WLL} that $P^J_n(\Lambda)=P_n(\Lambda)$, where $P^J_{n}$ and $P_{n}$ are the $n$-th  Sweedler power maps of $H^J$ and $H$ respectively. Moreover, $P_n(\Lambda)$ is a central element of $H$ (see \cite[Proposition 3.1]{WLL}). Since $\Delta^J(\Lambda)=Q_J^{-1}\Lambda_{(1)}\otimes\Lambda_{(2)}Q_J,$ it follows that
\begin{equation}gin{equation}\label{equ006}\mathbf{u}^J:=S^J(\Lambda_{(2)}Q_J)Q_J^{-1}\Lambda_{(1)}=Q_J^{-1}S(Q_J)\mathbf{u},\end{equation}
where $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}.$ For $H$-module $V$ with the character $\chi_V$, we denote
$V^J$ the same as $V$ as  $\mathbbm{k}$-linear space but thought of as an $H^J$-module. Then the character of $V^J$ is also $\chi_V$. For any $n\in\mathbb{Z}$, we have
\begin{equation}gin{align*}
\nu_n(V^J)&=\chi_V((\mathbf{u}^J)^{-1}P_n^J(\Lambda))\\&=\chi_V(\mathbf{u}^{-1}S(Q_J^{-1})Q_JP_n^J(\Lambda))\ \ \text{by}\ (\ref{equ006})\\
&=\chi_V(\mathbf{u}^{-1}S(Q_J^{-1})Q_JP_n(\Lambda))\\&=\chi_V(\mathbf{u}^{-1}S^2(J^{-(2)})S(J^{-(1)})S(J^{(1)})J^{(2)}P_n(\Lambda))\\
&=\chi_V(J^{-(2)}\mathbf{u}^{-1}S(J^{-(1)})S(J^{(1)})J^{(2)}P_n(\Lambda))\\&=\chi_V(\mathbf{u}^{-1}S(J^{-(1)})S(J^{(1)})J^{(2)}P_n(\Lambda)J^{-(2)})\\
&=\chi_V(\mathbf{u}^{-1}S(J^{-(1)})S(J^{(1)})J^{(2)}J^{-(2)}P_n(\Lambda))\\&=\chi_V(\mathbf{u}^{-1}P_n(\Lambda))\\
&=\nu_n(V).
\end{align*}
We complete the proof.
\qed

We are now ready to state the main result which says that higher FS indicators are gauge invariants of the tensor category Rep$(H)$.
\begin{equation}gin{thm}
Let $H$ and $H'$ be two finite dimensional semisimple Hopf algebras over the field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. If $\mathcal{F}:\text{Rep}(H)\rightarrow\text{Rep}(H')$ is an equivalence of tensor categories, then $\nu_n(V)=\nu_n(\mathcal{F}(V))$ for any $n\in\mathbb{Z}$ and any finite dimensional representation $V$ of $H$.
\end{thm}
\proof
Since the  $\mathbbm{k}$-linear equivalence $\mathcal{F}:\text{Rep}(H)\rightarrow\text{Rep}(H')$
is a tensor equivalence, it follows from \cite[Theorem 2.2]{NS1} that $H$ and $H'$ are gauge equivalent in the sense that there exist
a twist $J$ of $H$  such that $H'$ is isomorphic to $H^J$ as bialgebras. Let $\sigma:H'\rightarrow H^J$ be such an isomorphism. Then $\sigma$ is
automatically a Hopf algebra isomorphism. The isomorphism $\sigma$ induces a $\mathbbm{k}$-linear equivalence $(-)^\sigma:\text{Rep}(H)\rightarrow\text{Rep}(H')$ as follows:  for any finite dimensional $H$-module $V$, $V^{\sigma}=V$ as $\mathbbm{k}$-linear space with the $H'$-module action given by $h'v=\sigma(h')v$ for $h'\in H'$, $v\in V$, and $f^\sigma=f$ for any morphism $f$ in Rep$(H)$.
Moreover, the equivalence  $\mathcal{F}$ is naturally isomorphic to the $\mathbbm{k}$-linear equivalence $(-)^\sigma$ (see \cite[Theorem 1.1]{KMN}).
Therefore, $$\nu_n(\mathcal{F}(V))=\nu_n(V^{\sigma}).$$
Let $\Lambda'$ be a nonzero integral of $H'$ and $S'$ the antipode of $H'$. Note that the map $\sigma: H'\rightarrow H^J$ is a Hopf algebra isomorphism. It follows that $\sigma(\Lambda')=\Lambda$, which is a nonzero integral of $H^J$ and $\sigma(P'_n(\Lambda'))=P^J_n(\Lambda)$, where $P'_n$ and $P^J_n$ are the $n$-th Sweedler power maps of $H'$
and $H^J$ respectively. In particular,
$$\sigma((\mathbf{u}')^{-1}P'_n(\Lambda'))
=(\mathbf{u}^J)^{-1}P^J_n(\Lambda),
$$
where $\mathbf{u}'=S'(\Lambda'_{(2)})\Lambda'_{(1)}$ and $\mathbf{u}^J=S^J(\Lambda_{\langle2\rangle})\Lambda_{\langle1\rangle}$.
We have
\begin{equation}gin{align*}
\nu_n(V^{\sigma})&=\chi_{V^{\sigma}}((\mathbf{u}')^{-1}P'_n(\Lambda'))\\
&=\chi_{V^{J}}(\sigma((\mathbf{u}')^{-1}P'_n(\Lambda')))\\
&=\chi_{V^{J}}((\mathbf{u}^J)^{-1}P^J_n(\Lambda))\\
&=\nu_n(V^J)\\
&=\nu_n(V),
\end{align*}where the last equality follows from Proposition \ref{thm5}.
We conclude that $\nu_n(\mathcal{F}(V))=\nu_n(V)$ for any $n\in\mathbb{Z}$ and any finite dimensional representation $V$ of $H$.  \qed

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