8.3 Representation Theory of the Symmetric Group

Representation Theory of the Symmetric Group is a powerful tool for understanding permutations and symmetry. It uses Young diagrams and tableaux to visualize partitions and construct irreducible representations called Specht modules.

These ideas connect permutations, partitions, and representations in surprising ways. The branching rule and Young's rule show how representations decompose, revealing deep connections between combinatorics and group theory.

Young diagrams and tableaux

Partitions and Young diagrams

Top images from around the web for Partitions and Young diagrams

• 

  Young symmetrizer - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Young symmetrizer - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Young symmetrizer - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Young symmetrizer - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Young symmetrizer - Wikipedia, the free encyclopedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Partitions and Young diagrams

• 

  Young symmetrizer - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Young symmetrizer - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Young symmetrizer - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Young symmetrizer - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Young symmetrizer - Wikipedia, the free encyclopedia View original

  Is this image relevant?

1 of 3

• A partition of a positive integer $n$ is a sequence of positive integers $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_k)$ satisfying $\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_k$ and $\lambda_1 + \lambda_2 + \ldots + \lambda_k = n$
• The Young diagram of a partition $\lambda$ represents the partition visually as a left-justified array of boxes with $\lambda_i$ boxes in the $i$-th row
  • For example, the partition $(4, 2, 1)$ of $7$ has the Young diagram: \square & \square & \square & \square \\ \square & \square \\ \square \end{array}$$
• The conjugate partition $\lambda'$ is obtained by transposing the Young diagram, i.e., reflecting it across the main diagonal
  • For the partition $(4, 2, 1)$, the conjugate partition is $(3, 2, 1, 1)$ with the Young diagram: \square & \square & \square \\ \square & \square \\ \square \\ \square \end{array}$$

Young tableaux and the Robinson-Schensted correspondence

• A Young tableau is a Young diagram filled with positive integers that are strictly increasing in columns and weakly increasing in rows
  • For example, a Young tableau of shape $(4, 2, 1)$: 1 & 2 & 4 & 7 \\ 3 & 5 \\ 6 \end{array}$$
• A standard Young tableau has the numbers $1, 2, \ldots, n$ each appearing exactly once
  • For the partition $(4, 2, 1)$, a standard Young tableau: 1 & 3 & 5 & 7 \\ 2 & 6 \\ 4 \end{array}$$
• The Robinson-Schensted correspondence establishes a bijection between permutations in $S_n$ and pairs of standard Young tableaux of the same shape
  • This correspondence has important applications in representation theory and combinatorics, as it relates permutations to Young tableaux and partitions

Specht modules for symmetric groups

Definition and construction of Specht modules

• For each partition $\lambda$ of $n$, there is an associated Specht module $S^\lambda$, a submodule of the permutation module $M^\lambda$ corresponding to the Young subgroup $S_\lambda$
• The Specht module $S^\lambda$ is generated by a special element called a polytabloid, which is a linear combination of tabloids (equivalence classes of Young tableaux under row equivalence) with coefficients given by the sign of the permutation
  • For example, for the partition $(2, 1)$, the polytabloid $e_t$ associated with the Young tableau $t = \begin{array}{cc} 1 & 2 \\ 3 \end{array}$ is: $e_t = \begin{array}{cc} 1 & 2 \\ 3 \end{array} - \begin{array}{cc} 2 & 1 \\ 3 \end{array}$
• The action of the symmetric group $S_n$ on the Specht module $S^\lambda$ is defined by permuting the entries of the Young tableaux and extending linearly to the whole module

Irreducibility and completeness of Specht modules

• The Specht modules $S^\lambda$, as $\lambda$ ranges over all partitions of $n$, form a complete set of irreducible representations of the symmetric group $S_n$ over a field of characteristic $0$
  • This means that every irreducible representation of $S_n$ is isomorphic to a Specht module, and Specht modules corresponding to different partitions are non-isomorphic
• The dimension of the Specht module $S^\lambda$ is given by the number of standard Young tableaux of shape $\lambda$, which can be computed using the Hook Length Formula
  • For example, the dimension of the Specht module $S^{(3, 2)}$ is $5$, as there are $5$ standard Young tableaux of shape $(3, 2)$: 1 & 2 & 3 \\ 4 & 5 \end{array}, \quad \begin{array}{ccc} 1 & 2 & 4 \\ 3 & 5 \end{array}, \quad \begin{array}{ccc} 1 & 2 & 5 \\ 3 & 4 \end{array}, \quad \begin{array}{ccc} 1 & 3 & 4 \\ 2 & 5 \end{array}, \quad \begin{array}{ccc} 1 & 3 & 5 \\ 2 & 4 \end{array}$$

Branching rule for representations

Statement and interpretation of the branching rule

• The branching rule describes how an irreducible representation of $S_n$ decomposes when restricted to the subgroup $S_{n-1}$
• If $S^\lambda$ is the Specht module corresponding to the partition $\lambda$ of $n$, then the restricted module $\operatorname{Res}^{S_n}_{S_{n-1}}(S^\lambda)$ is isomorphic to the direct sum of Specht modules $S^\mu$, where $\mu$ ranges over all partitions of $n-1$ obtained by removing a single box from the Young diagram of $\lambda$
  • For example, the branching rule for the Specht module $S^{(3, 2)}$: $\operatorname{Res}^{S_5}_{S_4}(S^{(3, 2)}) \cong S^{(2, 2)} \oplus S^{(3, 1)}$ as the partitions $(2, 2)$ and $(3, 1)$ are obtained by removing a single box from the Young diagram of $(3, 2)$

Proof and applications of the branching rule

• The proof of the branching rule relies on the combinatorial properties of Young tableaux and the action of the symmetric group on the basis of the Specht modules
  • The key idea is to show that the restricted Specht module has a basis indexed by standard Young tableaux of shape $\mu$, where $\mu$ is obtained by removing a box from $\lambda$
• The branching rule is a powerful tool for studying the representation theory of symmetric groups and constructing irreducible representations recursively
  • It allows for the computation of the character table of $S_n$ using the Murnaghan-Nakayama rule, which expresses the character values in terms of ribbon tableaux
  • The branching rule also has applications in the study of the cohomology of flag varieties and the combinatorics of symmetric functions

Permutation modules decomposition

Definition and properties of permutation modules

• A permutation module $M^\lambda$ is a module induced from the trivial representation of a Young subgroup $S_\lambda$ to the whole symmetric group $S_n$
  • The Young subgroup $S_\lambda$ is the direct product of symmetric groups $S_{\lambda_1} \times S_{\lambda_2} \times \cdots \times S_{\lambda_k}$, where $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_k)$ is a partition of $n$
• The permutation module $M^\lambda$ has a basis indexed by the cosets of $S_\lambda$ in $S_n$, and the action of $S_n$ on $M^\lambda$ is given by permuting the cosets
  • For example, for the partition $(2, 1)$, the permutation module $M^{(2, 1)}$ has a basis $\{v_1, v_2, v_3\}$ corresponding to the cosets: $v_1 \leftrightarrow S_{(2, 1)}, \quad v_2 \leftrightarrow (1 \; 3)S_{(2, 1)}, \quad v_3 \leftrightarrow (2 \; 3)S_{(2, 1)}$

Young's rule and Kostka numbers

• Young's rule states that the permutation module $M^\lambda$ decomposes into a direct sum of Specht modules $S^\mu$, where $\mu$ ranges over all partitions of $n$ that dominate the partition $\lambda$ (i.e., $\mu_1 + \cdots + \mu_i \geq \lambda_1 + \cdots + \lambda_i$ for all $i$)
  • For example, the decomposition of the permutation module $M^{(2, 1)}$: $M^{(2, 1)} \cong S^{(3)} \oplus S^{(2, 1)}$ as the partitions $(3)$ and $(2, 1)$ dominate the partition $(2, 1)$
• The multiplicities of the Specht modules in the decomposition of $M^\lambda$ are given by the Kostka numbers, which count the number of semistandard Young tableaux of shape $\mu$ and content $\lambda$
  • A semistandard Young tableau is a Young tableau filled with positive integers that are weakly increasing in rows and strictly increasing in columns
  • The content of a tableau is the composition $(\alpha_1, \alpha_2, \ldots)$, where $\alpha_i$ is the number of occurrences of $i$ in the tableau
  • For example, the Kostka number $K_{(2, 1), (2, 1)} = 2$, as there are two semistandard Young tableaux of shape $(2, 1)$ and content $(2, 1)$: 1 & 1 \\ 2 \end{array}, \quad \begin{array}{cc} 1 & 2 \\ 2 \end{array}$$
• The decomposition of permutation modules into irreducible Specht modules is a fundamental result in the representation theory of symmetric groups and has applications in various areas of mathematics, such as algebraic combinatorics and invariant theory
  • It provides a way to construct irreducible representations of $S_n$ from the more easily understood permutation modules
  • The Kostka numbers and their generalizations, such as Littlewood-Richardson coefficients, play a crucial role in the study of symmetric functions and the representation theory of general linear groups