Hook of a Young diagram ($H_{i,j}(\lambda )$)

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 arm consists of the cells strictly to the right of $(i,j)$ in its row of the Young diagram; its length is $a_{\lambda }(i,j)={\lambda }_i-j$. It excludes the anchor cell. At $(1,2)$ in $(3,2)$ the arm contains only $(1,3)$, so its length is one. Together with the leg of a Young-diagram cell and the anchor, it forms the hook of a Young diagram.

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.

The hook length of cell $(i,j)$ in the Young diagram of a partition of an integer $\lambda$ is the number of cells in its Young-diagram hook:Here ${\lambda }^{\prime }$ is the conjugate partition. There is one hook length for every cell, so the multiset of hook lengths has cardinality $|\lambda |$.

For $\lambda =(3,2)$ and ${\lambda }^{\prime }=(2,2,1)$, compute $h_{1,1}=(3-1)+(2-1)+1=4$ and $h_{1,2}=(3-2)+(2-1)+1=3$. Continuing cell by cell gives the hook graph of a partitionor the row array $[[4,3,1],[2,1]]$. Its hook product is $4\cdot 3\cdot 1\cdot 2\cdot 1=24$. A cell has hook length one exactly when it is a Removable node of a Young diagram: it has no cell to its right or below. Here these are $(1,3)$ and $(2,2)$. Transposing the Young diagram gives the array $[[4,2],[3,1],[1]]$ with the same hook product; this explains geometrically why conjugate shapes have equal counts in the hook-length formula.

The hook graph is the Young diagram with the hook length written in each box. It displays the hook product and thus the hook-length formula for the corresponding Specht module. This use of the name is recorded in Frame, Robinson and Thrall's original hook-graph paper; it refers to a labeled array, not a directed hook-walk graph.

The product of all hook lengths in a Young diagram is the denominator in the hook-length formula, and the scalar in the quasi-idempotence identity for its Young symmetrizer.

For $(i,j)\in Y(\lambda )$, the integers from one through $h_{i,j}$ split as the disjoint unionTracing the southeast boundary of the hook records the first set at horizontal steps and the second at vertical steps.

If a partition has a hook of length $ef$, it has a hook of length $e$. In a beta set, the first hook is a bead-gap pair at distance $ef$; along the arithmetic progression with step $e$, some consecutive pair changes from a bead to a gap.

A rim hook, or border strip, is a connected skew Young diagram containing no $2\times 2$ square. Its leg length is one less than its number of occupied rows.

For a type $\alpha$, each successive difference is a rim hook of size ${\alpha }_i$; zero parts give empty differences of height zero. Label its cells by $i$. Such fillings are weakly increasing along rows and columns, and the height is the sum of the occupied strips' numbers of rows minus one. Empty parts are omitted from the associated product of positive power sums; the finite-variable degree-zero power sum is not inserted.