[proofplan] We use the [inverse symmetry of the Robinson-Schensted correspondence](/theorems/8437): passing from a permutation to its inverse interchanges the insertion and recording tableaux. Once this tableau identity is known, the cell statement is only a translation of the definitions of the left and right tableau cells. The key point is to track the convention: $L_T$ is indexed by the recording tableau $Q$, while $R_T$ is indexed by the insertion tableau $P$. [/proofplan] [step:Apply inverse symmetry of RSK to exchange insertion and recording tableaux] Let $n \in \mathbb N$, and let $w \in S_n$. The permutation $w^{-1}$ is again an element of $S_n$, so the RSK tableaux $P(w^{-1})$ and $Q(w^{-1})$ are defined. By the RSK inverse-symmetry theorem [citetheorem:8437], applied to the permutation $w$, one has \begin{align*} P(w^{-1}) = Q(w). \end{align*} \begin{align*} Q(w^{-1}) = P(w). \end{align*} [guided] Let $w \in S_n$. The RSK correspondence assigns to $w$ a pair \begin{align*} \operatorname{RSK}(w)=(P(w),Q(w)), \end{align*} where $P(w)$ is the insertion tableau and $Q(w)$ is the recording tableau. Since inversion is a map $S_n \to S_n$, the inverse permutation $w^{-1}$ also has an RSK pair \begin{align*} \operatorname{RSK}(w^{-1})=(P(w^{-1}),Q(w^{-1})). \end{align*} The inverse-symmetry theorem for RSK states that inversion interchanges these two tableaux. Applying [citetheorem:8437] to the permutation $w$ gives exactly \begin{align*} P(w^{-1}) = Q(w). \end{align*} \begin{align*} Q(w^{-1}) = P(w). \end{align*} This is the tableau-level content of the theorem. [/guided] [/step] [step:Translate the tableau identities into the cell equivalence] Let $T$ be a standard Young tableau with $n$ boxes. By definition, \begin{align*} L_T = \{u \in S_n : Q(u)=T\}, \qquad R_T = \{u \in S_n : P(u)=T\}. \end{align*} Using the first identity from the previous step, we obtain \begin{align*} w \in L_T \iff Q(w)=T \iff P(w^{-1})=T \iff w^{-1}\in R_T. \end{align*} This proves the stated equivalence and completes the proof. [/step]