Young-Tableau
A Young tableau or Young diagram ( after Alfred Young) is a graphical tool to the representation theory of the symmetric group Sn. Each Young tableau is determined by a certain number of cells ( usually symbolized by squares) determined that from top to bottom and left-aligned so that the number in each new row does not increase.
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 of each line and is used for compact description of its structure. In the examples shown arise following partitions: a) ( 4,2,2,1 ) b ) (1) c ) ( 1,1,1,1 ) and d) (4). The order n of the Tableaux denotes the number of all cells. The number of valid tableaux of order n can be given by the partition function.
- 2.1 The outer tensor 2.1.1 example
- 2.2.1 Example
Properties
The most important relationships between the irreducible representations of Sn and the Young tableaux of order n are outlined here.
Young schema and the projectors of the irreducible representations
A Young diagram is a Young tableau whose n cells are initially assigned randomly with the numbers from 1 to n. Examples of Young's schemes:
A) b) c) d) Now operators of these schemes are formed. The lines form in the scheme, the basis for the formation of an operator P. For each line are formed and summed permutations of all combinations of cells indices. The resulting sums of permutations are multiplied. Analogously form the columns in the schema, the basis for the formation of an operator Q. For each column are formed from all combinations of column indices permutations and summed. In the summation, but with a negative sign is used when the permutation is odd. The resulting sums of permutations are multiplied.
Example:
Here is true ( in the cycle notation)
And
Default schema
A standard pattern is a Young's scheme in which the numbering of the cells is performed such that the numbers are larger in each column from top to bottom, and each line from left to right.
Examples of standard schemes:
Important phrases
For the schemes can show the following
- The operator R = PQ is a scalar multiple of a projector. That is: RR = R k, where k is a constant other than 0, which also sets the normalization of R (R / k is a normalized projector). In the following, should always be meant the normalized projectors.
- The projectors of the different schemes are orthogonal tableaux: Ri Rj = 0
- The projectors to all schemes the same tableaux are not linearly independent - but such to any standard schemes of a given tableau. These can then be a system of orthogonal projectors Rik Rim = 0 construct.
- The system of all projectors Rik for all i Tableaux with all possible k standard schemes is complete, that is, the sum of all ( normalized) Rik is 1
- The number of orthogonal projectors Rik (at standard schemes ), so can be constructed from tableaux of order n, and the sum of the dimensions of the irreducible representations of Sn is equal.
Thus, the Rik, the projectors of the irreducible representations of Sn.
The outer tensor product of representations of symmetric groups: Littlewood - Richardson coefficients
The outer tensor
Two representations of two (generally different ) symmetric groups and you can "link" to a representation of the symmetric group, the so-called outer tensor product of these two representations. The exact definition of this presentation is as follows:
For any two permutations, and we define the " outer product " as the permutation of the set, which maps to each and every on maps. So put it clearly is the permutation that such acts on the first n numbers and the last m numbers such as ( a shifted by n permutation ) acts.
We can use the group as a subgroup of View ( by virtue of the embedding ).
For each representation of V and W of each presentation, we now define the outer tensor product of V and W as the display ( in this case on canonical way a representation of the group: the group is active on the first Tensoranden, while the group to the second acts Tensoranden ).
The outer product of the Sn associated permutations of Si, which act on the indices 1 to i, where the permutations Sj, i 1 to i j act on indicators and together describe permutations of Si j. This raises the question, in which irreducible representations of the Si j decays, the outer product of an irreducible representation of Si and Sj. In the following, the outer product is represented by the symbol ''.
Example
As an example we choose. Be the trivial representation of (ie the one-dimensional vector space, where each element of the identity acts ) and is the alternating representation ( also Signum representation or signature representation called ) of ( ie the one-dimensional vector space on which every even permutation as an identity and every odd permutation as a point reflection in the origin acts ). Then a one-dimensional representation of the group, and the outer product of, and is a six-dimensional representation of.
The question of the decomposition
Now the question arises, as the outer tensor product of two irreducible representations can be decomposed into irreducible representations ( this tensor is itself rarely irreducible, but by the theorem of Maschke it decomposes into a direct sum of irreducible representations ). Since the irreducible representations of (up to isomorphism ) uniquely the Young tableaux of order corresponding to k, so we can ask the following question:
Let T and S two Young tableaux of orders n and m, respectively. Let V and W, the irreducible representations of or belonging to these Young tableaux. The outer product of v and w is then a representation of, and thus a direct sum of irreducible representations. These irreducible representations in turn correspond to Young tableaux of order n m. Which Young tableaux are these? We briefly review
To say, the Young's tableaux are the irreducible representations of in which decomposes the outer product of v and w. It may also happen that one and the same table multiple times under the Young Tableaux - namely, if an irreducible representation occurs more than once in the decomposition of the outer product of V and W. Sometimes it holds in this case, these same tableaux together (rather than to write so if is ). This is the sum of a sum of pairwise distinct Young tableaux with coefficients - these coefficients are called Littlewood -Richardson coefficients.
The question now is how to determine the basis of T and S, the Young tableaux. There are different answers to this question; they are generally called Littlewood -Richardson rules (after Dudley Littlewood and AR Richardson ). We give below such a rule, which is recursive (there are explicit rules, which, however, a tedious combinatorial formulation have ).
Example
First, an example: Let T and S the Young tableaux
T = [ ] [] and S = [] []. To T or S belonging irreducible representations then V and W are the trivial preparation of ( as V) and the alternating display of (as W). We are thus in the example above, where we have found that the exterior product of V and W is a 6- dimensional representation of. It can be observed (eg with character theory ) that this representation can be written as a direct sum, where the irreducible representation of the Young tableau
[ ] [ ] [] [ ], and the irreducible representation of the Young tableau
[ ] [] [ ] [ ] is. So we can write:
T ( X) S = [] [] ( X ) [] = [] [ ] [] (X ) [] [ ] [ ] [ ] [] [] where we P ( X) Q for write.
A calculation method for T ( X) S
Now let the Young tableaux T and S given. We want the summands in the decomposition determine (in the above example, you could quite easily do this by hand, especially with character theory, but for larger Tableaux this will quickly become very tedious ).
The so-called Pieri rule takes care of this in the special case when the tableau S consists of only one line: In this case, the sum of all Young tableaux resulting from the Young tableau T by appending a total of m new cells (where m is the order of S ), with a maximum of a new cell per column.
For example (the asterisk is only used as a guide for the assignment of the cells):
[ ] [] ( X) [* ] [* ] = [ ] [ ] [* ] [* ] [] [ ] [* ] [] [ ] [* ] [] [ ] [ ] [ ] [ ] [ *] [ ] [ ] [* ] [*] [ *] A combination as
[ ] [] [ ] [*] [*] does not occur in the development, because in it the first column contains two cells added [* ].
To form the outer product (X) between any Tableaux you disassembled first one of the two tableaux in an alternating sum of outer products of one-line tableaux according to the following rule: Do we have a tableau of the form ( i, j, ..., n, m ) in front of us, then we calculate the outer product (i, j, ..., n ) (X) (m). We get a sum of tableaux, including our initial tableau (i, j, ..., n, m), but also some more tableaux. These further tableaux are now deducted:
(i, j, ..., n, m) = (i, j, ..., n ) (X ) (M ) - ( some more tableaux ).
On the resulting sum, the procedure is applied recursively. This recursion always comes to an end, because with every step tableaux arise, who in the last line at least one cell less.
For example (the asterisk is only used as a guide for the assignment of the cells):
[ ] [] = [] [] ( X) [* ] [* ] - [ ] [ ] [* ] [* ] - [ ] [ ] [* ] [ ] [ ] [* ] = [] [] ( X) [* ] [* ] - [ ] [ ] [* ] [* ] - ([] [ ] [] ( X) [* ] - [ ] [ ] [ ] [* ] ) After this decomposition, one can perform the actual multiplication by exploiting the associativity of the outer product and using the Pieri rule. An application of the outer product can be found in the decomposition of the tensor representation of a many-body system.
Warning
The outer tensor product of two representations V and W are two symmetric groups and is not to be confused with the inner tensor product of two representations V and W one and the same symmetric group. The latter is ( as I said ) is defined only 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 of a representation is ). The decomposition of this inner tensor product into irreducible representations is still a lot more difficult than that of the outer tensor product. Instead of the Littlewood - Richardson coefficients here called Kronecker coefficients come into play.
Importance
The use of Young tableaux is diverse. They serve
- To determine the dimensionality of the irreducible representations of the symmetric group
- For the construction of projectors on the subspaces of the irreducible representations of the symmetric group
- As an aid in the proof of propositions related to the symmetric group
- Decomposition of the outer product into its irreducible components
In addition, for example, in elementary particle physics using the technique of Young tableaux is a decomposition of the tensor of Mehrteilchensystemen possible. Among other things, they were used to elucidate the quark structure of hadrons. Quarks were initially not directly observed by high-energy scattering experiments, but had to be inferred from the scheme of the realized as representations of the underlying group composite particles initially.