Glossary

5.2 Character tables and their construction

2 min read•july 25, 2024

are powerful tools in representation theory, encapsulating a group's structure and behavior. They're built systematically, using and irreducible characters to reveal key and relationships.

These tables serve as group fingerprints, helping identify isomorphisms and structural similarities. By analyzing , conjugacy class sizes, and , we gain deep insights into a group's representations and overall nature.

Character Tables and Their Construction

Construction of character tables

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  • Construct character tables for through systematic steps
    1. Identify conjugacy classes of the group (, )
    2. Determine size of each conjugacy class ()
    3. Find number of irreducible characters (equal to number of conjugacy classes)
    4. Calculate degrees of irreducible characters (divisors of )
    5. Use orthogonality relations to fill remaining entries (row and )
  • reflect group structure
    • Rows correspond to irreducible characters, showing representation behavior
    • Columns correspond to conjugacy classes, grouping elements with similar properties
    • First row always (all 1's), representing
    • First column represents character degrees, indicating
  • Orthogonality relations ensure consistency and provide computational tools
    • : ∑g∈Gχi(g)χj(g)‾=∣G∣δij\sum_{g \in G} \chi_i(g) \overline{\chi_j(g)} = |G| \delta_{ij} ( of distinct characters)
    • Column orthogonality: ∑χ∈Irr(G)χ(g)χ(h)‾=∣CG(g)∣δgh\sum_{\chi \in Irr(G)} \chi(g) \overline{\chi(h)} = |C_G(g)| \delta_{gh} (relation between conjugacy classes)

Irreducible characters from tables

  • Number of irreducible characters determined by table structure
    • Equals number of conjugacy classes, reflecting group's complexity
    • Matches number of rows in character table, providing complete representation set
  • relates group order to conjugacy class sizes
    • ∣G∣=∑i=1k∣Ci∣|G| = \sum_{i=1}^k |C_i|, where CiC_i are conjugacy classes (partition of group elements)
  • Sum of squares of degrees equals group order
    • ∑χ∈Irr(G)χ(1)2=∣G∣\sum_{\chi \in Irr(G)} \chi(1)^2 = |G| ()
    • Provides check for completeness of character table

Conjugacy classes in tables

  • Conjugacy classes form columns in character table
    • Each column represents elements with same conjugation behavior
  • Size of conjugacy class CiC_i calculated using character values
    • ∣Ci∣=∣G∣χ1(gi)|C_i| = \frac{|G|}{\chi_1(g_i)}, where gig_i is class representative
    • Utilizes first row (principal character) of table
  • Class sizes invariant under isomorphism, preserving group structure
  • Center elements form singleton conjugacy classes, indicating commutativity

Isomorphism class from tables

  • Character tables serve as
    • Identical tables suggest possible isomorphism between groups
  • Key features for comparison reveal structural similarities
    • Group order (sum of squared degrees)
    • Number of conjugacy classes (complexity of group)
    • Degrees of irreducible characters (dimensions of representations)
    • (behavior of representations)
  • Techniques for distinguishing non-isomorphic groups
    • Check for abelian vs non-abelian structure (number of linear characters)
    • Examine number of linear characters ()
    • Analyze (normal subgroups)
  • Limitations of character table analysis
    • Non-isomorphic groups can share character tables (e.g., D8D_8 and Q8Q_8)
    • Additional group-theoretic information may be needed for definitive classification (subgroup structure, automorphisms)

Key Terms to Review (25)

Abelian Groups: Abelian groups are mathematical structures that consist of a set equipped with an operation that combines any two elements to form a third element, following specific properties. They are characterized by the property that the group operation is commutative, meaning that the order in which two elements are combined does not affect the result. This property is crucial in understanding representations, particularly irreducible representations, as it influences how symmetries can be represented in a linear format and affects the construction and interpretation of character tables.
Burnside's Theorem: Burnside's Theorem provides a powerful method for counting the number of distinct objects under group actions, specifically by relating the number of orbits to the average number of points fixed by the group elements. This theorem lays the groundwork for understanding how symmetry operates in various contexts, revealing insights into character theory, representation analysis, and finite group classifications.
Character degrees: Character degrees are the dimensions of the irreducible representations of a finite group, which can be found in the character table. These degrees provide important information about the structure of the group and help to classify its representations, revealing relationships between different representations and aiding in understanding the group's properties.
Character Table Properties: Character table properties refer to the specific attributes and relationships within a character table, which is a square matrix that encodes information about the irreducible representations of a finite group. These properties include orthogonality relations, the ability to recover group structure, and connections to conjugacy classes, making character tables essential for understanding the representation theory of groups.
Character Tables: Character tables are mathematical tools used in representation theory to encapsulate the information of irreducible representations of a finite group. They summarize how each irreducible representation corresponds to the conjugacy classes of the group through character values, which are the traces of the corresponding matrices. This organization makes it easier to analyze and classify representations, revealing deep connections between group structure and representation theory.
Class equation: The class equation is a fundamental formula in group theory that relates the size of a finite group to the sizes of its conjugacy classes and the centralizer of the identity element. It provides a powerful tool to analyze the structure of groups, particularly in understanding their irreducible representations and character tables. By expressing the order of a group in terms of the sizes of conjugacy classes, it reveals important information about how representations can be constructed and classified.
Column orthogonality: Column orthogonality refers to the property of a matrix where its columns are mutually orthogonal, meaning the inner product of any two distinct columns is zero. This concept is crucial in representation theory as it ensures that different representations do not interfere with each other, which helps in simplifying the analysis and construction of character tables.
Conjugacy Classes: Conjugacy classes are subsets of a group formed by grouping elements that are related through conjugation, meaning if one element can be transformed into another via an inner automorphism. Each conjugacy class represents a distinct behavior of elements in a group and plays a crucial role in understanding the structure of groups, especially when constructing character tables, analyzing irreducible representations, applying Frobenius reciprocity, and utilizing Mackey's theorem.
Cyclic subgroups: A cyclic subgroup is a subset of a group that can be generated by a single element, meaning every element in the subgroup can be expressed as powers (or multiples) of that generator. This concept is fundamental in group theory as it highlights the structure within groups and helps in understanding character tables and induction processes, especially since each cyclic subgroup has its own characters that contribute to the overall character table of a group.
Dimension of representations: The dimension of representations refers to the size of a representation, specifically the number of basis elements in the vector space associated with a group representation. This concept is key in understanding how groups can act on vector spaces, as it directly relates to the complexity and structure of these representations. The dimension can influence the properties of the representation, including its reducibility and its relationship to character theory.
Distribution of character values: The distribution of character values refers to the way in which the values of irreducible characters are arranged for the elements of a group, typically organized in a character table. This concept is crucial for understanding how representations of a group behave under conjugation and highlights the relationship between group elements and their symmetries. It connects deeply with various group properties, helping to identify subgroup structures and analyze the representation theory more effectively.
Finite groups: Finite groups are mathematical structures consisting of a set with a finite number of elements along with a binary operation that satisfies the group axioms: closure, associativity, the identity element, and inverses. They are essential in representation theory because they help us understand how abstract algebraic structures can be represented through linear transformations and matrices. The properties of finite groups play a significant role in character theory, where we analyze homomorphisms and derive character tables to study the representations of these groups more deeply.
Group Order: Group order refers to the total number of elements present in a group, which is a fundamental concept in group theory. Understanding group order is essential as it impacts various properties of groups, including their structure and behavior. It plays a critical role in several important results, such as Burnside's theorem, where it helps in determining the number of orbits of a group action. Additionally, the order of a group aids in interpreting orthogonality relations and constructing character tables, making it a cornerstone of representation theory.
Group properties: Group properties are characteristics that define how a group operates and the structures it forms. These properties help in understanding the group's structure, behavior, and symmetry, which are crucial for analyzing group representations and character tables. Key properties include whether a group is abelian, finite, or simple, among others, as these influence how we can construct character tables and interpret their significance.
Identity representation: Identity representation refers to the simplest and most fundamental representation of a group in the context of a group action. It is essentially the trivial representation where every group element acts as the identity transformation, meaning that each group element maps every vector in the representation space to itself. This representation serves as a baseline for understanding more complex representations and plays a critical role in character tables.
Inner Product: An inner product is a mathematical operation that takes two vectors and produces a scalar, serving as a generalization of the dot product. It captures geometric notions like length and angle between vectors, which are crucial when analyzing representations and their decompositions. Inner products enable us to define orthogonality, leading to insights about how representations can be broken down into simpler, irreducible components and how characters behave within the framework of group theory.
Isomorphism class: An isomorphism class refers to a set of mathematical structures that can be transformed into one another through an isomorphism, meaning they share the same structure and properties. This concept is essential in representation theory, as it helps to group together representations of a group that are fundamentally the same despite being realized in different ways. Understanding isomorphism classes enables one to categorize and analyze the character tables and constructions related to groups efficiently.
Isomorphism invariants: Isomorphism invariants are properties or characteristics of algebraic structures that remain unchanged under isomorphisms, meaning they can be used to determine whether two structures are essentially the same. In the context of character tables, these invariants help identify distinct representations and categorize groups by their characters. By examining these invariants, one can gain insight into the structure of a group and its representations.
Kernel of characters: The kernel of characters refers to the set of irreducible representations of a group that correspond to the trivial character. This concept plays a critical role in understanding how group representations behave under direct limits and extensions, and it helps identify the structural properties of a group. The kernel provides insight into the relationships between different representations and can help classify the group's characters within the context of their character tables.
Normal subgroups: Normal subgroups are special types of subgroups within a group where the subgroup is invariant under conjugation by any element of the group. This means that for any element in the group and any element in the normal subgroup, the result of conjugating the subgroup element by the group element will still be in the subgroup. This property is crucial for defining quotient groups and has deep implications in both representation theory and the study of character tables.
One-dimensional representations: One-dimensional representations are homomorphisms from a group to the multiplicative group of non-zero complex numbers, typically denoted as $$ ext{Hom}(G, imes \mathbb{C}^\times)$$. This means that each element of the group is associated with a complex number in a way that respects the group operation. These representations play a crucial role in representation theory by allowing us to analyze groups through their characters and provide insights into the structure of groups.
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.
Orthogonality Relations: Orthogonality relations are mathematical statements that describe how different representations and their corresponding characters interact with one another, often resulting in specific inner product relationships that provide insights into the structure of a group. These relations show that the inner product of characters associated with different irreducible representations is zero, reflecting the idea that distinct representations do not overlap in a certain way. Understanding these relations is crucial for analyzing the properties of irreducible representations, constructing character tables, and applying character theory to finite group theory.
Row orthogonality: Row orthogonality refers to the property in representation theory where the rows of a character table are orthogonal with respect to a specific inner product defined on the space of class functions. This property ensures that the inner product of two different rows (representations) in the character table results in zero, indicating that they are linearly independent. This concept is crucial for understanding the structure of representations and their decompositions in the context of finite groups.
Trivial character: The trivial character is a specific type of character in representation theory that maps every group element to the number one. This means it assigns a constant value of one to every element in the group. The trivial character is essential because it provides a baseline for understanding other characters, and its properties help establish foundational results in the study of representations and character tables.
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