group generated by the automorphism defined by $\sigma(x_i)=\xi x_i$ for $i=1,\dots,m$, where $\xi$ is an $n$-th primitive root of unity. If $\operatorname{char}\kk\nmid n$, then $\Pty(R,G)=\operatorname{GKdim}(R)$.

For any elements $a_1,\dots,a_n\in R_1$, we see $\sigma(a_i)=\xi a_i$ for $i=1,\dots,n$. By Lemma \[lem2\], we have $a_1a_2\cdots a_n\in \mfr(R,G)$. Note that $R_k=(R_1)^k$ for all $k\ge1$. Then we see $R_n\subseteq \mfr(R,G)$, and hence $R_i\subseteq \mfr(R,G)$ for all $i\ge n$. Therefore the pertinency algebra $\mathfrak{R}(R,G)$ is finite dimensional. Hence $\Pty(R,G)=\operatorname{GKdim}(R)$.

Let $R=\kk_{-1}[x_1,\dots,x_n]$ $(n\ge2)$ be the skew-symmetric algebra. Let $\sigma$ be the automorphism of $R$ defined by $\sigma(x_i)=x_{i+1}$ for $i=1,\dots,n-1$ and $\sigma(x_n)=x_1$. Let $G\leq \operatorname{Aut}(R)$ be the subgroup generated by $\sigma$. The pertinency $\Pty(R,G)$ was computed in [@BHZ]. We give an alternative computation in this paper.

We may choose another set of generators for $R$ so that $\sigma$ acts on the generators diagonally. Let $\xi$ be an $n$-th primitive root of unity. For $j=1,\dots,n$, let $y_j=\sum_{i=1}^n\xi^{ji}x_i$. Then $\{y_1,\dots,y_n\}$ is a set of linear independent generators of $R$, and $\sigma(y_j)=\xi^{-j}y_i$ for $j=1,\dots,n$. For $1\leq k\leq n-1$, we have $$\sum_{i=0}^{n-1}y_j^i\sigma^k(y_j^{n-i})=\sum_{i=0}^{n-1}\xi^{-jk(n-i)}y_j^n.$$ If $\gcd(j,n)=1$, then $n\nmid jk$. In this case, $\sum_{i=0}^{n-1}\xi^{-jk(n-i)}=0$, and hence $\sum_{i=0}^{n-1}y_j^i\sigma^k(y_j^{n-i})=0$, which in turn implies that the sequences $(1,y_j,\dots,y_j^{n-1})$ and $(y_j^n,y_j^{n-1},\dots,y_j)$ are pertinent under the $G$-action. Hence we have 