Editing Young tableau (section)

==Applications in representation theory==
{{see also|Representation theory of the symmetric group}}
Young diagrams are in one-to-one correspondence with [[irreducible representation]]s of the [[symmetric group]] over the [[complex number]]s. They provide a convenient way of specifying the [[Young symmetrizer]]s from which the [[representation theory of the symmetric group|irreducible representations]] are built. Many facts about a representation can be deduced from the corresponding diagram. Below, we describe two examples: determining the dimension of a representation and restricted representations. In both cases, we will see that some properties of a representation can be determined by using just its diagram. Young tableaux are involved in the use of the symmetric group in
quantum chemistry studies of atoms, molecules and solids.<ref>Philip R. Bunker and Per Jensen (1998) ''Molecular Symmetry and Spectroscopy'', 
2nd ed.  NRC Research Press, Ottawa [https://volumesdirect.com/products/molecular-symmetry-and-spectroscopy?_pos=1&_sid=ed0cc0319&_ss=r]
pp.198-202.{{ISBN|9780660196282}}</ref><ref>R.Pauncz (1995) ''The Symmetric Group in Quantum Chemistry'',
CRC Press, Boca Raton, Florida</ref>

Young diagrams also parametrize the irreducible polynomial representations of the [[general linear group]] {{math|''GL''<sub>''n''</sub>}} (when they have at most {{mvar|''n''}} nonempty rows), or the irreducible representations of the [[special linear group]] {{math|''SL''<sub>''n''</sub>}} (when they have at most {{math|''n'' − 1}} nonempty rows), or the irreducible complex representations of the [[special unitary group]] {{math|''SU''<sub>''n''</sub>}} (again when they have at most {{math|''n'' − 1}} nonempty rows). In these cases semistandard tableaux with entries up to {{mvar|''n''}} play a central role, rather than standard tableaux; in particular it is the number of those tableaux that determines the dimension of the representation.

===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>

===Restricted representations===

A representation of the symmetric group on {{mvar|''n''}} elements, {{math|''S''<sub>''n''</sub>}} is also a representation of the symmetric group on {{math|''n'' − 1}} elements, {{math|''S''<sub>''n''−1</sub>}}. However, an irreducible representation of {{math|''S''<sub>''n''</sub>}} may not be irreducible for {{math|''S''<sub>''n''−1</sub>}}. Instead, it may be a [[direct sum of representations|direct sum]] of several representations that are irreducible for {{math|''S''<sub>''n''−1</sub>}}. These representations are then called the factors of the [[restricted representation]] (see also [[induced representation]]).

The question of determining this decomposition of the restricted representation of a given irreducible representation of ''S''<sub>''n''</sub>, corresponding to a partition {{mvar|''λ''}} of {{mvar|''n''}}, is answered as follows. One forms the set of all Young diagrams that can be obtained from the diagram of shape {{mvar|''λ''}} by removing just one box (which must be at the end both of its row and of its column); the restricted representation then decomposes as a direct sum of the irreducible representations of {{math|''S''<sub>''n''−1</sub>}} corresponding to those diagrams, each occurring exactly once in the sum.