Conjugate partition
= Conjugate partition
{title2=$\lambda'$}
The conjugate partition exchanges rows and columns in a <Young diagram>: the cell $(i,j)$ becomes $(j,i)$. Equivalently,
$$
\lambda'_j=\#\{i:\lambda_i\geq j\},\qquad
Y(\lambda')=\{(j,i):(i,j)\in Y(\lambda)\}.
$$
For $\lambda=(3,2)$, the three column lengths are $2,2,1$, so $\lambda'=(2,2,1)$, 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 $h_{j,i}(\lambda')=h_{i,j}(\lambda)$; 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 https://doi.org/10.4153/CJM-1954-030-1[Frame, Robinson and Thrall's 1954 paper]{ref}.