Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 160 / 3 / a

Trace the southeast boundary of the Hook of a Young diagram based at $(i,j)$. At each horizontal boundary step record the hook length $h_{i,y}$ of the cell in row $i$ above that step. At each vertical step ending beside row $x$, record $h_{i,j}-h_{x,j}$. Starting at the northeast end and moving to the southwest end, these records increase by one from $1$ to $h_{i,j}$; horizontal and vertical steps are disjoint and account for every step. Therefore the Hook-interval decomposition at a Young-diagram cell is

The supplied row-hook formula isConsequently $h\in H_i(\lambda )$ exactly when $h_i-h\ge 0$ and $h_i-h$ is not one of $h_{i+1},\dots ,h_m$. Since $h_i-h\le h_i<h_j$ for $j<i$, this is equivalent to $h_i-h\notin X$. We have proved the hook criterion in a beta set

If $ef$ is a hook length, the beta-set interpretation gives a bead at some position $b$ and a gap at $b-ef$. In the finite progressionthe first position is occupied and the last is empty. Some consecutive pair is therefore a bead followed by a gap. Their distance is $e$, so the criterion gives a hook of length $e$. This proves the divisor closure of hook lengths.