Editing Young tableau (section)

===Dimension of a representation===
{{Main|Hook length formula}}
{{Plain image with caption|
Image:Hook length for 541 partition.svg|caption=''Hook-lengths'' of the boxes for the partition 10&nbsp;=&nbsp;5&nbsp;+&nbsp;4&nbsp;+&nbsp;1}}

The dimension of the irreducible representation {{math|{{pi}}<sub>''λ''</sub>}} of the symmetric group {{math|''S''<sub>''n''</sub>}} corresponding to a partition {{mvar|''λ''}} of {{mvar|''n''}} 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 the [[hook length formula]].

A '''hook length''' {{math|hook(''x'')}} of a box {{mvar|''x''}} in Young diagram {{math|''Y''(''λ'')}} of shape {{mvar|''λ''}} 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 {{math|''n''!}} divided by the product of the hook lengths of all boxes in the diagram of the representation:

:<math>\dim\pi_\lambda = \frac{n!}{\prod_{x \in Y(\lambda)} \operatorname{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>\dim\pi_\lambda = \frac{10!}{7\cdot5\cdot 4 \cdot 3\cdot 1\cdot 5\cdot 3\cdot 2\cdot 1\cdot1} = 288.</math>

Similarly, the dimension of the irreducible representation {{math|''W''(''λ'')}} of {{math|GL<sub>''r''</sub>}} corresponding to the partition ''λ'' of ''n'' (with at most ''r'' parts) is the number of semistandard Young tableaux of shape ''λ'' (containing only the entries from 1 to ''r''), which is given by the hook-length formula:

: <math>\dim W(\lambda) = \prod_{(i,j) \in Y(\lambda)} \frac{r+j-i}{\operatorname{hook}(i,j)},</math>

where the index ''i'' gives the row and ''j'' the column of a box.<ref>{{cite book|author=Predrag Cvitanović |year=2008 |title=Group Theory: Birdtracks, Lie's, and Exceptional Groups | publisher=Princeton University Press | url=http://birdtracks.eu/|author-link=Predrag Cvitanović }}, eq. 9.28 and appendix B.4</ref> For instance, for the partition (5,4,1) we get as dimension of the corresponding irreducible representation of {{math|GL<sub>7</sub>}} (traversing the boxes by rows):

:<math>\dim W(\lambda) = \frac{7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 6\cdot 7\cdot 8\cdot 9\cdot 5}{7\cdot5\cdot 4 \cdot 3\cdot 1\cdot 5\cdot 3\cdot 2\cdot 1\cdot1} = 66 528.</math>