Glossary

9.3 Representations of symmetric and alternating groups

2 min read•july 25, 2024

representations are a cornerstone of representation theory. They use and to visualize and construct irreducible representations, while character theory and help calculate and relate them.

representations build on symmetric group theory, using restriction and branching. They involve , , and . These concepts have wide-ranging applications in algebra and combinatorics.

Symmetric Group Representations

Irreducible representations of symmetric group

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  • Young diagrams visually represent integer partitions and correspond to irreducible representations (5 + 3 + 2 partition of 10)
  • Specht modules constructed using tabloids and polytabloids provide basis for irreducible representations
  • Character theory employs and for calculations
  • Branching rules describe relationship between representations of SnS_n and Sn−1S_{n-1}
  • determines of irreducible representations n!∏(i,j)∈λhij\frac{n!}{\prod_{(i,j)\in \lambda} h_{ij}}

Symmetric group representations vs integer partitions

  • exists between partitions and irreducible representations
  • graphically represents partitions and connects to representation theory
  • on partitions establishes partial ordering of representations
  • relate to dual representations (transpose Young diagram)
  • involves composition of representations and partitions

Alternating Group Representations

Irreducible representations of alternating group

  • Derived from symmetric group representations through restriction and branching rules
  • Self-conjugate partitions lead to splitting of representations
  • Spin representations arise from double covers of alternating groups
  • Character theory utilizes induced characters from symmetric group
  • Clifford theory applied to analyze alternating group representations

Applications in algebra and combinatorics

  • connects representations of symmetric and general linear groups
  • Invariant theory studies polynomial invariants of symmetric groups
  • Combinatorial applications include Robinson-Schensted-Knuth correspondence and
  • Algebraic applications involve of and symmetric functions
  • Group actions and counting utilize Burnside's lemma and
  • examines FI-modules and asymptotic behavior of representations

Key Terms to Review (24)

Alternating Group: The alternating group, denoted as A_n, is the group of all even permutations of n elements, which can be thought of as the set of all permutations that can be expressed as a product of an even number of transpositions. This group plays a crucial role in both group theory and representation theory, particularly in understanding the structure and characteristics of symmetric groups, which include both even and odd permutations. The alternating groups are simple for n ≥ 5, meaning they have no nontrivial normal subgroups, which makes them fundamental in the study of abstract algebra and the classification of finite simple groups.
Bijection: A bijection is a specific type of function that establishes a one-to-one correspondence between two sets, meaning each element in the first set is paired with a unique element in the second set, and vice versa. This property ensures that the function is both injective (no two elements map to the same element) and surjective (every element in the target set is mapped by some element from the domain). Understanding bijections is crucial when working with representations of groups, particularly in relating group elements to vector spaces.
Branching rules: Branching rules are principles that describe how representations of a group can be restricted to its subgroups or induced from them. These rules help understand the behavior of representations as one moves between a group and its subgroups, allowing for a systematic approach to analyzing how larger representations can break down into simpler components when viewed through the lens of smaller groups.
Clifford Theory: Clifford Theory is a fundamental result in representation theory that connects representations of a group with those of its subgroups, particularly highlighting how induction and restriction functors behave in this context. It provides a framework for understanding how the representations of a group can be constructed from the representations of its subgroups, which plays a significant role in analyzing more complex structures like symmetric and alternating groups. By emphasizing the relationships between different groups, Clifford Theory aids in various applications including the analysis of character theory and decomposition of representations.
Conjugate Partitions: Conjugate partitions are a specific way of transforming a partition of an integer into another partition by flipping the rows and columns in its Ferrers diagram. This concept is significant in the study of symmetric and alternating groups as it helps to describe the irreducible representations of these groups, connecting the structure of partitions to the representation theory.
Decomposition: Decomposition refers to the process of breaking down a representation into simpler components or irreducible representations. This is an essential concept that highlights how complex structures can often be understood by examining their fundamental parts, connecting to properties such as the uniqueness and simplicity of these components in various mathematical frameworks.
Dimension: Dimension in representation theory refers to the size of a vector space associated with a representation, specifically the number of basis vectors needed to span that space. This concept is crucial as it relates to understanding the structure of representations, particularly how they can be decomposed and analyzed, influencing topics such as irreducibility and induced representations.
Dominance order: The dominance order is a way to compare partitions of a set, particularly in the context of symmetric groups and their representations. It establishes a hierarchy among partitions based on their sizes and structure, indicating how one partition can be 'greater' or 'less' than another in terms of their composition. This ordering is crucial for understanding how different representations relate to each other, especially when analyzing characters and their dimensions.
Frobenius Formula: The Frobenius formula provides a method for calculating the characters of representations of symmetric groups by relating them to the cycle types of permutations. This formula plays a crucial role in understanding the structure of symmetric and alternating groups, allowing for the determination of characters by summing over contributions from the different cycle types. It highlights how these representations can be analyzed through combinatorial techniques, linking representation theory with group theory.
Hook length formula: The hook length formula is a mathematical expression used to calculate the dimension of a representation of the symmetric group corresponding to a given partition of a positive integer. It connects combinatorial concepts with representation theory, offering insights into how symmetric and alternating groups act on vector spaces associated with partitions, especially when determining characters and dimensions.
Irreducible Representation: An irreducible representation is a linear representation of a group that cannot be decomposed into smaller, non-trivial representations. This concept is crucial in understanding how groups act on vector spaces, as irreducible representations form the building blocks from which all representations can be constructed, similar to prime numbers in arithmetic.
Littlewood-Richardson Rule: The Littlewood-Richardson Rule provides a combinatorial method to compute the coefficients appearing in the expansion of the product of two Schur functions in terms of a basis of Schur functions. This rule is crucial for understanding how representations of certain groups can be decomposed, particularly in contexts involving tensor products, symmetric and alternating groups, and highest weight theory.
Murnaghan-Nakayama Rule: The Murnaghan-Nakayama Rule is a combinatorial formula used to compute the characters of the symmetric group based on the cycle type of permutations. This rule connects the representation theory of symmetric groups with combinatorial aspects by providing a method to derive character values from certain tableaux associated with partitions. It highlights the relationship between representation theory and algebraic combinatorics, emphasizing how different representations can be constructed from standard Young tableaux and their properties.
Orbit-stabilizer theorem: The orbit-stabilizer theorem is a fundamental result in group theory that relates the size of a group acting on a set to the sizes of orbits and stabilizers. Specifically, it states that for a group acting on a set, the size of the orbit of an element multiplied by the size of its stabilizer equals the size of the group. This theorem is crucial for understanding how group actions partition sets into orbits and provides insights into counting arguments and representation theory.
Plethysm: Plethysm is a mathematical operation that combines two representations of symmetric groups to produce a new representation. This operation is significant in the context of representation theory because it enables the study of how representations interact and can be decomposed into simpler components. It serves as a bridge between combinatorial identities and algebraic properties, particularly in understanding the structure of representations of symmetric and alternating groups.
Representation Stability: Representation stability refers to a phenomenon in representation theory where the dimensions of the spaces of representations for a given group, as it varies with respect to some parameter (like degree or size), exhibit a predictable and stable pattern. This concept highlights that as the group grows or changes, the representations maintain certain structural similarities, often leading to insights about their reducibility, equivalence, and how they relate to other groups, especially in specific applications like Frobenius reciprocity.
Schur-Weyl Duality: Schur-Weyl duality is a powerful concept in representation theory that describes the relationship between the representations of the symmetric group and those of a general linear group acting on a tensor space. This duality reveals how these two groups interact with each other through their respective representations, leading to a decomposition of tensor products and an understanding of how symmetric and alternating groups manifest in algebraic varieties.
Self-conjugate partitions: Self-conjugate partitions are specific types of partitions of an integer where the partition is equal to its own conjugate. This means that if you visualize the partition as a diagram, it remains unchanged when reflected across its main diagonal. They play an important role in the representation theory of symmetric groups, especially in understanding the structure of the irreducible representations of these groups.
Specht modules: Specht modules are a family of representations of the symmetric group, constructed using the theory of Young tableaux. They arise from partitioning a set and are instrumental in studying the representation theory of symmetric and alternating groups, as they provide a basis for the modular representation theory and allow for the computation of characters of these groups.
Spin Representations: Spin representations are specific types of representations of a group that involve the action of the group on a vector space that reflects the intrinsic angular momentum (spin) of quantum particles. These representations capture the behavior of particles with half-integer spin and are essential for understanding the structure of quantum mechanics and particle physics. Spin representations can be irreducible, meaning they cannot be decomposed into simpler representations, which ties into fundamental concepts in representation theory and plays a crucial role in the study of symmetric and alternating groups.
Symmetric group: The symmetric group, denoted as $$S_n$$, is the group of all permutations of a finite set of $$n$$ elements, capturing the essence of rearranging objects. This group is fundamental in understanding how groups act on sets, with its elements representing all possible ways to rearrange the members of the set, leading to various applications in algebra and combinatorics.
Tensor products: Tensor products are a way to combine two vector spaces or modules into a new one, allowing for a broader understanding of linear transformations and their representations. They capture how two algebraic structures interact and are crucial in areas like representation theory and algebraic geometry, providing insights into the relationships between different representations. This concept is particularly important when considering the Frobenius reciprocity theorem and the representations of symmetric and alternating groups.
Young diagrams: Young diagrams are graphical representations of partitions, typically used to illustrate the representation theory of symmetric groups. They consist of rows of boxes, with each row representing a part of the partition, and are instrumental in understanding how representations can be constructed and decomposed, particularly in the context of tensor products and the representations of symmetric and alternating groups.
Young's Lattice: Young's Lattice is a combinatorial structure that organizes partitions of a natural number into a partially ordered set, where each element represents a partition. This lattice provides a way to visualize the relationships between different partitions and plays an important role in understanding the representation theory of symmetric and alternating groups, especially in how representations can be decomposed into irreducible components.
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