Consider a 2D system with a non-anomalous finite non-abelian global symmetry $G$, for example $$G = S_3=\{e,a,a^2,b,ab,a^2b\}$$ with $a^3=b^2=1$. One expects the local operators charged under the symmetry $S_3$ should transform under some representation of $S_3$ and one can decompose into three irreducible representations $I,P,E$. Here $I$ is the trivial representation, $P$ is the sign representation characterized by $P(b)=-1$ and $E$ is a 2-dimensional representation given by
\begin{equation} E(a) = \left[ \begin{array}{cc} \omega&0 \newline 0&\omega^2 \end{array} \right],\quad E(b) = \left[ \begin{array}{cc} 0&1 \newline 1&0 \end{array} \right], \end{equation}
where $\omega$ is cube root of unity. If we gauge the $S_3$-symmetry, I learned that the dual symmetry should be a symmetry category Rep($S^3$) which is non-invertible and the generators of the symmetry are given by the irreducible representations $I,P,E$ satisfying
\begin{equation} P^2 = I,\quad E^2 = I + P + E,\quad PE = EP =E. \end{equation}
My question is, what's the corresponding "representation" under the representation symmetry Rep($S_3$)? For example, how does the local operators in the new theory transform under the action of Rep($S_3$)?
I guess this question has been answered somewhere but I cannot find a straightforward answer. It will be great if anyone can give me some references on that.
