Sigma involutions associated with parafermion vertex operator algebra $K({\mathfrak{s}\mathfrak{l}}_2,k)$

An irreducible module for the parafermion vertex operator algebra $K({\mathfrak{s}\mathfrak{l}}_2,k)$ is said to be of $\sigma$-type if an automorphism of the fusion algebra of $K({\mathfrak{s}\mathfrak{l}}_2,k)$ of order $k$ is trivial on it. For any integer $k\ge 3$, we show that there exists an automorphism of order $2$ of the subalgebra of the fusion algebra of $K(\mathfrak{sl}_2,k)^{\langle \theta \rangle}$spannedbytheirreducibledirectsummandsof$\sigma$-type irreducible $K({\mathfrak{s}\mathfrak{l}}_2,k)$-modules, where $\theta$ is an involution of $K({\mathfrak{s}\mathfrak{l}}_2,k)$. We discuss some examples of such an automorphism as well.