 Solution

The conjugate partition is ${\lambda }_j^{\prime }=\#\{i:{\lambda }_i\ge j\}$, obtained by transposing the Young diagram. Transposition sends a standard Young tableau $T$ to a standard tableau $T^{\prime }$ and negates every Content of a Young-diagram cell.

Use the Young orthogonal form. For an admissible pair $R=s_{i}T$ and axial distance $d=c_T(i+1)-c_T(i)$, the action on $(w_T,w_R)$ isIn the transposed pair it is $A_{-d}$, whereas in the tensor product of group representations with the sign representation it is $-A_d$. Their diagonal entries agree, and their off-diagonal entries differ by a sign. Fix a reference tableau and let ${ϵ}_T$ be the sign of the unique label permutation from it to $T$. For every admissible swap ${ϵ}_R=-{ϵ}_T$, sointertwines these two actions. In a nonadmissible pair, transposition exchanges row and column and changes the scalar from $+1$ to $-1$ or conversely, also agreeing with the sign twist. Since adjacent transpositions generate $S_n$, this is an isomorphism:The parity choice is globally well defined because the label permutation is unique; no arbitrary edge-by-edge phase choices are required.