= Hook of a Young diagram
{title2=$H_{i,j}(\lambda)$}
= Hooks of a Young diagram
{synonym}
= Young-diagram hook
{c}
{synonym}
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:
$$
H_{i,j}(\lambda)=\{(i,j)\}
\cup\{(i,k):j<k\leq\lambda_i\}
\cup\{(k,j):i<k\leq\lambda'_j\}.
$$
Here $\lambda'$ 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 https://doi.org/10.4153/CJM-1954-030-1[Frame, Robinson and Thrall's original paper]{ref}.
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