Conjugate partition (${\lambda }^{\prime }$)

The conjugate partition exchanges rows and columns in a Young diagram: the cell $(i,j)$ becomes $(j,i)$. Equivalently,For $\lambda =(3,2)$, the three column lengths are $2,2,1$, so ${\lambda }^{\prime }=(2,2,1)$, as shown in Young diagram. Conjugating twice returns the original partition of an integer, and the number of cells stays five. Transposition also preserves hook lengths, with $h_{j,i}({\lambda }^{\prime })=h_{i,j}(\lambda )$; consequently the hook-length formula gives equal dimensions for the two Specht modules over the complex numbers. This row-column symmetry is already used in §1 of ^[ref].