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[[Image:Hook length for 541 partition.svg|Hook-lengths of the boxes for the partition ''10=5+4+1'']]<br>''Hook lengths''</div>
The dimension of the irreducible representation <math>\pi_\lambda\,\!</math> of the symmetric group <math>S_n</math>, corresponding to a partition <math>\lambda\,\!</math> of <math>n</math>, is equal to the number of different standard Young tableaux that can be obtained from the diagram of the representation. This number can be calculated by ''hook-length formula''.
A hook length <math>\operatorname{hook}(x)</math> of a box <math>x</math> in Young diagram <math>Y(\lambda)\,\!</math> of shape <math>\lambda\,\!</math> is the number of boxes that are in the same row to the right of it plus those boxes in the same column below it, plus one (for the box itself). By the hook-length formula, the dimension of an irreducible representation is ''n''! divided by the product of the hook lengths of all boxes in the diagram of the representation:
:<math>{\rm dim} \, \pi_\lambda = \frac{n!}{\prod_{x \in Y(\lambda)} {\rm hook}(x)}.</math>
The figure on the right shows hook-lengths for all boxes in the diagram of the partition 10 = 5 + 4 + 1. Thus <math>{\rm dim} \, \pi_\lambda = \frac{10!}{1\cdot1\cdot 1 \cdot 2\cdot 3\cdot 3\cdot 4\cdot 5\cdot 5\cdot7} = 288 </math>.
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