8.4 Specht Modules and Young's Lattice

Specht modules and Young's lattice are key concepts in combinatorial representation theory. They provide a concrete way to understand the irreducible representations of symmetric groups using tableaux and diagrams. These tools connect abstract algebra to visual, countable structures.

The Littlewood-Richardson rule and Schur-Weyl duality further link representation theory to combinatorics. They show how tensor products decompose and how different groups' actions relate, bridging algebra and counting problems in surprising ways.

Specht modules from polytabloids

Constructing Specht modules

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• Specht modules are irreducible representations of the symmetric group constructed using polytabloids
• The Specht module corresponding to a partition $\lambda$ is the vector space spanned by all polytabloids obtained from standard Young tableaux of shape $\lambda$
• Polytabloids are constructed by applying the Young symmetrizer to a Young tableau
  • The Young symmetrizer is the element of the group algebra of the symmetric group obtained by symmetrizing over the rows and antisymmetrizing over the columns of a Young tableau
• Specht modules provide a concrete realization of the irreducible representations of the symmetric group

Young tableaux and standard Young tableaux

• A Young tableau is a filling of the boxes of a Young diagram with the numbers 1 through n, using each number exactly once
  • For example, a Young tableau of shape (3, 2) could be: \begin{ytableau} 1 & 3 & 5 \ 2 & 4 \end{ytableau}
• A standard Young tableau is a Young tableau in which the numbers increase along each row and down each column
  • The previous example is a standard Young tableau, but \begin{ytableau} 1 & 4 & 5 \ 2 & 3 \end{ytableau} is not standard
• The number of standard Young tableaux of shape $\lambda$ is equal to the dimension of the Specht module corresponding to $\lambda$
• Standard Young tableaux play a crucial role in the construction of Specht modules and the representation theory of the symmetric group

Littlewood-Richardson rule for tensor products

Tensor product decomposition

• The Littlewood-Richardson rule describes the decomposition of the tensor product of two irreducible representations of the general linear group into a direct sum of irreducible representations
• The multiplicity of an irreducible representation in the tensor product is given by the number of Littlewood-Richardson tableaux of the corresponding shape
• The rule provides a combinatorial method for computing tensor product multiplicities without explicitly constructing the representations
  • For example, the tensor product of the irreducible representations corresponding to the partitions (2, 1) and (1, 1) decomposes as: $V_{(2,1)} \otimes V_{(1,1)} \cong V_{(3,2)} \oplus V_{(3,1,1)} \oplus V_{(2,2,1)}$

Littlewood-Richardson tableaux and skew Young diagrams

• A skew Young diagram is the difference of two Young diagrams, where the smaller diagram is contained within the larger one
  • For example, the skew diagram (3, 2, 1) / (2, 1) is: \begin{ytableau} \none & \none & \square \ \none & \square \ \square \end{ytableau}
• A Littlewood-Richardson tableau is a skew Young tableau that satisfies certain conditions:
  • The word formed by reading the entries from right to left in each row, starting with the top row, is a lattice permutation (the content of any initial segment is dominated by the content of the previous initial segment)
  • The content of the skew tableau, which counts the number of occurrences of each number, matches the content of the Young diagram being added
• The number of Littlewood-Richardson tableaux of shape $\lambda / \mu$ with content $\nu$ gives the multiplicity of $V_\lambda$ in the tensor product $V_\mu \otimes V_\nu$

Structure of Young's lattice

Partial order and cover relations

• Young's lattice is a partially ordered set of all integer partitions, ordered by inclusion of Young diagrams
  • For example, (2, 1) ≤ (3, 2) because the Young diagram of (2, 1) is contained within the Young diagram of (3, 2)
• The cover relations in Young's lattice correspond to adding a single box to a Young diagram
  • (2, 1) is covered by (3, 1) and (2, 2) because these partitions can be obtained by adding a single box to (2, 1)
• The rank function in Young's lattice is given by the size of the partition, which is the sum of its parts
  • The rank of (3, 2, 1) is 3 + 2 + 1 = 6

Combinatorial properties and connections to representation theory

• Young's lattice has a unique minimal element, the empty partition (), and no maximal elements
• The number of partitions of n, denoted p(n), is the number of elements in Young's lattice at rank n
  • For example, p(4) = 5 because there are 5 partitions of 4: (4), (3, 1), (2, 2), (2, 1, 1), and (1, 1, 1, 1)
• The number of saturated chains from the empty partition to a partition $\lambda$ is equal to the dimension of the corresponding irreducible representation of the symmetric group
• The number of standard Young tableaux of shape $\lambda$ is equal to the number of maximal chains from the empty partition to $\lambda$ in Young's lattice
• These combinatorial properties highlight the deep connections between Young's lattice and the representation theory of the symmetric group

Schur-Weyl duality: Representations vs Combinatorics

Double centralizer theorem and Schur-Weyl duality

• Schur-Weyl duality is a fundamental relationship between the representation theory of the general linear group and the symmetric group
• The duality states that the actions of the general linear group GL(V) and the symmetric group S_n on the tensor power V^⊗n commute with each other, and their double centralizers are isomorphic
  • The centralizer of an action is the set of linear transformations that commute with the action
  • The double centralizer theorem states that the centralizer of the centralizer of an action is isomorphic to the original algebra acting on the space
• The irreducible representations of GL(V) that appear in the decomposition of V^⊗n are labeled by partitions of n, and their multiplicities are given by the dimensions of the corresponding irreducible representations of S_n

Schur functions and combinatorial interpretations

• The character of the irreducible representation of GL(V) labeled by a partition $\lambda$, evaluated on a matrix with eigenvalues x_1, ..., x_n, is given by the Schur function s_$\lambda$(x_1, ..., x_n)
• Schur functions are symmetric polynomials that have many combinatorial interpretations:
  • The Jacobi-Trudi formula expresses Schur functions as determinants of complete homogeneous symmetric functions: $s_\lambda = \det(h_{\lambda_i - i + j})$
  • The Littlewood-Richardson rule expresses the product of two Schur functions as a sum of Schur functions with coefficients given by Littlewood-Richardson numbers: $s_\mu s_\nu = \sum_\lambda c_{\mu\nu}^\lambda s_\lambda$
• Schur functions are closely related to the characters of irreducible representations of the general linear group and the symmetric group
• The combinatorial properties of Schur functions, such as the Littlewood-Richardson rule, have important applications in representation theory and algebraic combinatorics

Applications and connections

• Schur-Weyl duality provides a powerful tool for studying the combinatorics of Young tableaux, symmetric functions, and related objects, by relating them to the representation theory of the general linear and symmetric groups
• The duality allows for the transfer of results and techniques between representation theory and combinatorics
  • For example, the Littlewood-Richardson rule for multiplying Schur functions can be proved using the representation theory of the general linear group and Schur-Weyl duality
• Schur-Weyl duality has applications in various areas of mathematics and physics, such as:
  • Quantum mechanics and the study of angular momentum
  • Invariant theory and the study of polynomial invariants of the general linear group
  • Combinatorial representation theory and the study of representations of symmetric groups and related algebras