 Solution

ID: past-exam-of-the-lie-algebras-and-representation-theory-of-symmetric-groups-course-of-the-university-of-oxford/2009/b/4/1/solution

[λ] = {(i, j) : 1 \leq i \leq k, 1 \leq j \leq λ_{i}}

. The hook length of a box is

h (i, j) = λ_{i} - j + λ_{j}^{'} - i + 1,

counting the box itself, the boxes to its right in the same row, and the boxes below it in the same column. The hook-length formula is

f^{λ} = \frac{n !}{\prod _{(i, j) \in [λ]} h ( i , j )} .

The first implication in the equivalence is immediate: if every hook length is odd, no hook can have length two.

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