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In the English convention of Young diagram, the hook at a cell $(i,j)$ consists of that cell itself, all cells to its right in the same row, and all cells below it in the same column:Here ${\lambda }^{\prime }$ is the conjugate partition. The cells to the right form the arm of a Young-diagram cell, and those below form the leg of a Young-diagram cell. Both extend all the way to the edge of the Young diagram, rather than containing just the immediate neighbors. The three pieces are disjoint, so their total cardinality is one plus the two lengths.

For $(3,2)$, the highlighted Young-diagram hook is $H_{1,2}=\{(1,2),(1,3),(2,2)\}$, of hook length three. The hook at $(1,1)$ is $\{(1,1),(1,2),(1,3),(2,1)\}$, of hook length four: the southeast cell $(2,2)$ is excluded because it is in neither the anchor's row nor its column. These local shapes are useful because their hook lengths give the denominator of the hook-length formula. The hook terminology and its use to calculate representation degrees appear explicitly in §1 of ^[ref].

The leg consists of the cells strictly below $(i,j)$ in its column of the Young diagram; its length is $l_{\lambda }(i,j)={\lambda }_j^{\prime }-i$, where ${\lambda }^{\prime }$ is the conjugate partition. It excludes the anchor cell. At $(1,2)$ in $(3,2)$ the leg contains only $(2,2)$, so its length is one. Transposing the Young diagram exchanges the leg and the arm of a Young-diagram cell.