# Young tableux (representation theory)

• Ted123
In summary, Young tableaux are a visual and systematic tool used in representation theory to organize and understand irreducible representations of groups or algebras. They are constructed by arranging boxes in rows and columns, and the shape of the tableau corresponds to the symmetry properties of the representation. They can be used for any finite group or Lie algebra and are closely related to other important concepts in representation theory.
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## Homework Statement

Consider the irreducible representation $V$ in the symmetric group $S_5$ corresponding to the Young diagram (these are meant to be boxes): $$[\;\;][\;\;] \\ [\;\;][\;\;] \\ [\;\;]\;\;\;\;$$

(a) List all standard Young tableaux of the given shape (that is, list all the possible fillings of the boxes with $\{1,2,3,4,5\}$ such that the numbers increase along the rows and down the columns).

(b) Give the dimension of $V$.

(c) For any standard tableau $\Omega$ of the given shape, let $e_{\Omega}$ be the standard basis element of the representation $V$. Evaluate the action of the permutation $g=(23)(45)(23)(34)(23) \in S_5$ on the vector $e_{\Omega}$ where $\Omega =$ $$[\;1\;][\;4\;] \\ [\;2\;][\;5\;] \\ [\;3\;]\;\;\;\;\;\,$$

## The Attempt at a Solution

(a) There are 5 standard Young tableaux of the given shape.
(b) So dim(V)=5.
(c) The basis vector $e_{\Omega} = P_{\Omega}Q_{\Omega}$ where $P_\Omega \in \mathbb{C}S_5$ and $Q_{\Omega}\in\mathbb{C}S_5$ are the row symmetrizer and column antisymmetrizer of $\Omega$: $$P_{\Omega} = [e+(14)][e+(25)] = e + (25) + (14) + (14)(25) \\ Q_{\Omega} = [e - (12) - (13) - (23) + (123) + (132)][e-(45)]$$

The given permutation $g=(23)(45)(23)(34)(23)=(2543)$. Is there a quicker way to evaluate the action of this permutation on $e_{\Omega}$ then to work out $(2543)P_{\Omega}Q_{\Omega}$?

Last edited by a moderator:

Hello, thank you for your question. Here is my response:

(a) Yes, there are 5 standard Young tableaux of the given shape: [1][2] [3][4] [5], [1][3] [2][4] [5], [1][4] [2][3] [5], [1][5] [2][3] [4], [1][2] [3][5] [4].

(b) The dimension of V is indeed 5.

(c) Yes, there is a quicker way to evaluate the action of the given permutation on e_{\Omega}. Since the given permutation (23)(45)(23)(34)(23) is a product of disjoint cycles, we can evaluate its action on e_{\Omega} by simply permuting the numbers in each cycle. This means that (23)(45)(23)(34)(23) will map [1][2] [3][4] [5] to [2][3] [5][4] [1]. Therefore, the action of the given permutation on e_{\Omega} is [e+(23)(45)][e+(23)(45)] = [e+(45)(23)] = e.

## 1. What is the purpose of Young tableaux in representation theory?

Young tableaux are used in representation theory to provide a visual and systematic way of organizing and understanding the irreducible representations of a group or algebra. They also help to determine the branching rules for decomposing a representation into irreducible components.

## 2. How are Young tableaux constructed?

Young tableaux are constructed by arranging boxes in rows and columns, where the number of boxes in each row represents the number of times a particular irreducible representation appears in the decomposition of a larger representation. The columns represent the different irreducible representations in the decomposition.

## 3. What is the significance of the shape of a Young tableau?

The shape of a Young tableau corresponds to the symmetry properties of the irreducible representation it represents. This can provide information about the symmetry of a physical system or the symmetry of a mathematical object.

## 4. Can Young tableaux be used for any group or algebra?

Yes, Young tableaux can be used for any finite group or Lie algebra. They can also be extended to infinite groups and algebras, although the construction and properties may differ.

## 5. How are Young tableaux related to other concepts in representation theory?

Young tableaux are closely related to other important concepts in representation theory, such as character tables, Clebsch-Gordan coefficients, and tensor product decompositions. They provide a combinatorial approach to understanding these concepts and their connections.

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