Young tableau

A Young tableau or Young diagram (after Alfred Young ) is a graphical tool of the representation theory of the symmetrical group . Each Young tableau is determined by a certain number of cells (mostly symbolized by squares), which are arranged from top to bottom and left-justified in such a way that their number does not increase in each new line. ${\displaystyle S_{n}}$

Examples of valid Young tableaux:

a) [ ][ ][ ][ ]     b) [ ]      c) [ ]        d) [ ][ ][ ][ ]
   [ ][ ]                          [ ]
   [ ][ ]                          [ ]
   [ ]                             [ ]

Examples of not valid Young-Tableaux:

   [ ][ ][ ][ ]        [ ]
   [ ][ ]              [ ][ ]
   [ ][ ][ ]
   [ ]

The partition of a Young tableau is the enumeration of the number of cells in each row and is used for the compact description of its structure. The following partitions result in the examples shown: a) b) c) and d) . The order of the tableaux denotes the number of all cells. The number of valid tableaux with the order can be specified using the partition function . ${\displaystyle (4,2,2,1)}$ ${\displaystyle (1)}$ ${\displaystyle (1,1,1,1)}$ ${\displaystyle (4)}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle P(n)}$

properties

The most important connections between the irreducible representations of and the Young tableaux of the order are outlined here. ${\displaystyle S_{n}}$ ${\displaystyle n}$

Young's scheme and the projectors of the irreducible representations

A Young scheme is a Young tableau, the cells of which are initially filled with the numbers from to . Examples of Young schemes: ${\displaystyle n}$ ${\displaystyle 1}$ ${\displaystyle n}$

a) [3][7][6][5]     b) [1]     c) [1]     d) [3][4][2][1]
   [9][2]                         [2]
   [1][8]                         [3]
   [4]                            [4]

Now operators are formed from these schemes. The lines in the scheme form the basis for creating an operator . For each row, permutations are formed from all combinations of the cell indices and added up. The resulting sums of permutations are multiplied. Similarly, the columns in the schema form the basis for creating an operator . For each column, permutations are formed from all combinations of the column indices and added up. In the case of the summation, however, a negative sign is used if the permutation is odd . The resulting sums of permutations are multiplied. ${\displaystyle P}$ ${\displaystyle Q}$

Example:

   [3][1][6]
   [5][4]
   [2]

The following applies here (in cycle notation)

P = P ₁ P ₂ P ₃ = ( 1 + (3.1) + (3.6) + (1.6) + (3.1.6) + (1.3.6)) ( 1 + ( 5.4)) 1

and

Q = Q ₁ Q ₂ Q ₃ = ( 1 - (3.5) - (3.2) - (5.2) + (3.5.2) + (3.2.5)) ( 1 - ( 1,4)) 1

Standard scheme

A standard scheme is a Young scheme in which the numbering of the cells is carried out in such a way that the numbers increase in each column from top to bottom and in each row from left to right.

Examples of standard schemes:

  [1][3][6]     [1][3][5]     [1][2]     [1]
  [2][4]        [2][6]        [3]        [2]
  [5]           [4]                      [3]

Important sentences

The following can be shown for the schemes

• The operator is a scalar multiple of a projector. That means: where is a constant different from that which simultaneously defines the normalization for ( is normalized projector). In the following, the standardized projectors are always meant. ${\displaystyle R=PQ}$ ${\displaystyle RR=kR}$ ${\displaystyle k}$ ${\displaystyle 0}$ ${\displaystyle R}$ ${\displaystyle R/k}$
• The projectors to the schemes of different tableaux are orthogonal: . ${\displaystyle R_{i}R_{j}=0}$
• The projectors for all schemes of the same tableau are not linearly independent - but those for all possible standard schemes of a given tableau. A system of orthogonal projectors can then be constructed from these . ${\displaystyle R_{ik}R_{im}=0}$
• The system of all projectors to all tableaux with all possible standard schemes is complete, that is: the sum of all (standardized) is . ${\displaystyle R_{ik}}$ ${\displaystyle i}$ ${\displaystyle k}$ ${\displaystyle R_{ik}}$ ${\displaystyle 1}$
• The number of orthogonal projectors (for standard schemes) that can be constructed from tableaux of order and the sum of the dimensions of the irreducible representations is the same. ${\displaystyle R_{ik}}$ ${\displaystyle n}$ ${\displaystyle S_{n}}$

Thus the projectors are the irreducible representations of the . ${\displaystyle R_{ik}}$ ${\displaystyle S_{n}}$

The outer tensor product of representations of symmetric groups: Littlewood-Richardson coefficients

The outer tensor product

Two representations of two (generally different) symmetrical groups and can be "linked" to one representation of the symmetrical group , the so-called outer tensor product of these two representations. The exact definition of this representation is as follows: ${\displaystyle S_{n}}$ ${\displaystyle S_{m}}$ ${\displaystyle S_{n+m}}$

For any two permutations and we define the "outer product" than the permutation of the set , which each on maps and each on maps. So, clearly speaking, is the permutation, which acts like on the first numbers and like (a shifted permutation) on the last numbers . ${\displaystyle \sigma \in S_{n}}$ ${\displaystyle \pi \in S_{m}}$ ${\displaystyle \sigma \times \pi \in S_{n+m}}$ ${\displaystyle \left\{1,2,...,n+m\right\}}$ ${\displaystyle i\in \left\{1,2,...,n\right\}}$ ${\displaystyle \sigma \left(i\right)}$ ${\displaystyle j\in \left\{n+1,n+2,...,n+m\right\}}$ ${\displaystyle \pi \left(j-n\right)+n}$ ${\displaystyle \sigma \times \pi }$ ${\displaystyle n}$ ${\displaystyle \sigma }$ ${\displaystyle mk}$ ${\displaystyle n}$ ${\displaystyle \pi }$

We can view the group as a subgroup of (by embedding ). ${\displaystyle S_{n}\times S_{m}}$ ${\displaystyle S_{n+m}}$ ${\displaystyle S_{n}\times S_{m}\to S_{n+m},\ \left(\sigma ,\pi \right)\mapsto \sigma \times \pi }$

For each representation of and each representation of we now define the outer tensor product of and as the representation (here is a representation of the group in a canonical way : the group acts on the first tensor and the group acts on the second tensor). ${\displaystyle V}$ ${\displaystyle S_{n}}$ ${\displaystyle W}$ ${\displaystyle S_{m}}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle \mathrm {Ind} _{S_{n}\times S_{m}}^{S_{n+m}}\left(V\otimes W\right)}$ ${\displaystyle V\otimes W}$ ${\displaystyle S_{n}\times S_{m}}$ ${\displaystyle S_{n}}$ ${\displaystyle S_{m}}$

The outer product of the linked permutations of those that act on the indices up to with permutations of those that act on indices up to and together describe permutations of the . The question arises as to which irreducible representations the external product of an irreducible representation of and falls into. In the following, the outer product is shown with the symbol . ${\displaystyle S_{n}}$ ${\displaystyle S_{i}}$ ${\displaystyle 1}$ ${\displaystyle i}$ ${\displaystyle S_{j}}$ ${\displaystyle i+1}$ ${\displaystyle i+j}$ ${\displaystyle S_{i+j}}$ ${\displaystyle S_{i+j}}$ ${\displaystyle S_{i}}$ ${\displaystyle S_{j}}$ ${\displaystyle \otimes }$

example

We choose as an example . Let be the trivial representation of (i.e. the one-dimensional vector space on which each element of acts as an identity) and be the alternating representation (also called signum representation or signature representation) of (i.e. the one-dimensional vector space on which every even permutation acts as identity and every odd permutation acts as a point reflection at the origin). Then is a one-dimensional representation of the group , and the outer product of and is a six-dimensional representation of . ${\displaystyle n=m=2}$ ${\displaystyle V}$ ${\displaystyle S_{2}}$ ${\displaystyle S_{2}}$ ${\displaystyle W}$ ${\displaystyle S_{2}}$ ${\displaystyle V\otimes W}$ ${\displaystyle S_{2}\times S_{2}}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle \mathrm {Ind} _{S_{2}\times S_{2}}^{S_{4}}\left(V\otimes W\right)}$ ${\displaystyle S_{4}}$

The question of dismantling

Now the question arises of how the outer tensor product of two irreducible representations can be broken down into irreducible representations (this tensor product itself is only rarely irreducible, but according to Maschke's theorem it breaks down into a direct sum of irreducible representations). Since the irreducible representations of (except for isomorphism) clearly correspond to the Young tableaux of the order , we can ask the following question: ${\displaystyle S_{k}}$ ${\displaystyle k}$

Be and two Young tableaux of the orders respectively . Be and the irreducible representations of and that belong to these Young tableaux. The outer product of and is then a representation of , and thus a direct sum of irreducible representations of . These irreducible representations in turn correspond to Young tableaux of the order . Which Young Tableaux are these? We write briefly ${\displaystyle T}$ ${\displaystyle S}$ ${\displaystyle n}$ ${\displaystyle m}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle S_{n}}$ ${\displaystyle S_{m}}$ ${\displaystyle \mathrm {Ind} _{S_{n}\times S_{m}}^{S_{n+m}}\left(V\otimes W\right)}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle S_{n+m}}$ ${\displaystyle S_{n+m}}$ ${\displaystyle n+m}$

${\displaystyle T\otimes S=P_{1}+P_{2}+...+P_{l}}$

to say that the Young tableaux are to the irreducible representations of , into which the outer product of and is broken down. One and the same tableau can also appear several times among the Young tableaux - namely when an irreducible representation occurs several times in the decomposition of the outer product of and . Sometimes these same tableaux are summarized in this case (instead of writing , if is). This turns the sum into a sum of pairs of different Young tableaux with coefficients - these coefficients are called Littlewood-Richardson coefficients . ${\displaystyle P_{1},\ P_{2},\ ...,\ P_{l}}$ ${\displaystyle S_{n+m}}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle P_{1},\ P_{2},\ ...,\ P_{l}}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle P_{2}+P_{3}}$ ${\displaystyle 2P_{2}}$ ${\displaystyle P_{2}=P_{3}}$ ${\displaystyle P_{1}+P_{2}+...+P_{l}}$

The question now is how to determine the Young tableaux using and . There are different answers to this question; they are commonly referred to as the Littlewood-Richardson rules (after Dudley Littlewood and AR Richardson). In the following we give such a rule, which is recursive (there are also explicit rules, which, however, have a lengthy combinatorial formulation). ${\displaystyle T}$ ${\displaystyle S}$ ${\displaystyle P_{1},\ P_{2},\ ...,\ P_{l}}$

example

First an example: Be and the Young tableaux ${\displaystyle T}$ ${\displaystyle S}$

 T = [ ][ ]     und     S = [ ]
                            [ ] .

To or associated with irreducible representations and are then the trivial representation of (a ) and the alternating representation of (a ). So we are in the example above where we found that the outer product of and is a -dimensional representation of . One can determine (e.g. with character theory ) that this representation can be written as a direct sum , with the irreducible representation of being the Young tableau ${\displaystyle T}$ ${\displaystyle S}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle S_{2}}$ ${\displaystyle V}$ ${\displaystyle S_{2}}$ ${\displaystyle W}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle 6}$ ${\displaystyle S_{4}}$ ${\displaystyle U_{1}\oplus U_{2}}$ ${\displaystyle U_{1}}$ ${\displaystyle S_{4}}$

[ ][ ][ ]
[ ]

is, and the irreducible representation of to the Young tableau ${\displaystyle U_{2}}$ ${\displaystyle S_{4}}$

[ ][ ]
[ ]
[ ]

is. So we can write:

T (X) S = [ ][ ] (X) [ ] = [ ][ ][ ] (+) [ ][ ]
                     [ ]   [ ]           [ ]
                                         [ ]   ,

where we write P (X) Q for . ${\displaystyle P\otimes Q}$

A calculation method for T (X) S

Be they the Young tableaux and given. We want to determine the summands in the decomposition (in the above example this could still be done quite easily by hand, especially with character theory, but this quickly becomes very tedious for larger tableaux). ${\displaystyle T}$ ${\displaystyle S}$ ${\displaystyle P_{1},\ P_{2},\ ...,\ P_{l}}$ ${\displaystyle T\otimes S=P_{1}+P_{2}+...+P_{l}}$

The so-called Pieri rule does this in a special case when the tableau only consists of one row: In this case, the sum of all Youngtableaus that arise from the Youngtableau by adding a total of new cells (where the order is) is at most a new cell per column. ${\displaystyle S}$ ${\displaystyle T\otimes S}$ ${\displaystyle T}$ ${\displaystyle m}$ ${\displaystyle m}$ ${\displaystyle S}$

Example (the asterisk is only used as a guide when assigning the cells):

[ ][ ] (x) [*][*] = [ ][ ][*][*] + [ ][ ][*] + [ ][ ][*] + [ ][ ]
[ ]                 [ ]            [ ][*]      [ ]         [ ][*]
                                               [*]         [*]

A combination like

[ ][ ]
[ ]
[*]
[*]

does not occur in development because the first column contains two added cells [*].

To form the outer product between any tableaux, one of the two tableaux is first broken down into an alternating sum of outer products of one-line tableaux according to the following rule: If we have a tableau of the form in front of us, then we calculate the outer product . We get a total of tableaux, including our starting tableau , but also some other tableaux. These further tableaux are now deducted: ${\displaystyle (X)}$ ${\displaystyle (i,j,...,n,m)}$ ${\displaystyle (i,j,...,n)(X)(m)}$ ${\displaystyle (i,j,...,n,m)}$

${\displaystyle (i,j,...,n,m)=(i,j,...,n)(X)(m)-({\mbox{einige weitere Tableaux}})}$.

The procedure is applied recursively to the resulting sum. This recursion always comes to an end, because with each step tableaux are created that have at least one cell less in the last row.

Example (the asterisk is only used as a guide when assigning the cells):

[ ][ ] = [ ][ ] (X) [*][*] - [ ][ ][*][*] - [ ][ ][*]
[ ][ ]                                      [*]
       = [ ][ ] (X) [*][*] - [ ][ ][*][*] - ( [ ][ ][ ] (X) [*] - [ ][ ][ ][*] )

After this decomposition, the actual multiplication can be carried out using the associativity of the outer product and with the help of Pieri's rule. An application of the outer product can be found in the decomposition of the tensor representation of a many-particle system.

warning

The outer tensor product of two representations and two symmetric groups and should not be confused with the inner tensor product of two representations and one and the same symmetric group . The latter is (as said) only defined for two representations of the same symmetric group, and even then it differs from the outer tensor product (it is a representation of , while the outer tensor product is a representation of ). The decomposition of this inner tensor product into irreducible representations is even more difficult than that of the outer tensor product. Instead of the Littlewood-Richardson coefficients, so-called Kronecker coefficients come into play here. ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle S_{n}}$ ${\displaystyle S_{m}}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle S_{k}}$ ${\displaystyle S_{k}}$ ${\displaystyle S_{2k}}$

meaning

Young tableaux can be used in many ways. They serve among other things

• to determine the dimensionalities of the irreducible representations of the symmetric group
• for the construction of projectors on the subspaces of the irreducible representations of the symmetrical group
• as an aid in proving theorems related to the symmetric group
• to decompose the external product into its irreducible components

In addition, in elementary particle physics, the Young Tableaux technique enables a decomposition of the tensor representation of multi-particle systems. Among other things, they were used to elucidate the quark structure of hadrons . Initially, quarks were not observed directly through high-energy scattering experiments, but first had to be deduced from the systematics of the composite particles realized as representations of the underlying group.

See also

• Cycle type

literature

• William Fulton Young Tableaux. With applications to representation theory and geometry (= London Mathematical Society Student Texts. No. 35). Cambridge University Press, Cambridge et al. 1997, ISBN 0-521-56144-2 .

Web links

• Eric W. Weisstein : Young Tableau . In: MathWorld (English).