Glossary

2.2 Matrix representations and group algebras

2 min read•july 25, 2024

Matrix representations and group algebras are powerful tools in representation theory. They allow us to study abstract group structures using concrete linear algebra. These concepts bridge the gap between group theory and linear transformations.

Group algebras extend the idea of matrix representations. They provide a unified framework for studying all representations of a group simultaneously. This algebraic approach reveals deep connections between group structure and representation properties.

Matrix Representations

Matrix notation for linear representations

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  • Linear representations of groups map group elements to invertible linear transformations ( from G to GL(V))
  • Matrix representation chooses basis for vector space V, represents group elements as matrices
  • Properties preserve group structure: matrix multiplication corresponds to group operation, identity element becomes identity matrix
  • Dimension of representation determined by size of matrices (2x2, 3x3)
  • Examples: rotation matrices for cyclic groups, permutation matrices for symmetric groups, reflection matrices for dihedral groups

Group algebra construction

  • combines vector space over field F with group structure
  • Basis elements correspond to group elements, allowing formal linear combinations ∑g∈Gagg\sum_{g \in G} a_g g with ag∈Fa_g \in F
  • Dimension equals order of group (number of elements)
  • Multiplication defined by group operation, distributive over addition
  • Examples: group algebra of cyclic group C3C_3 has basis {e,g,g2}\{e, g, g^2\}, symmetric group S3S_3 has 6-dimensional algebra

Group Algebras and Matrix Representations

Group algebra vs matrix representations

  • Regular representation of group algebra acts on itself via left multiplication
  • Matrix representations correspond to algebra homomorphisms from group algebra
  • Group algebra elements expressed as linear combinations of representation matrices
  • decomposes group algebra into simple modules (irreducible representations)
  • theory connects traces of matrix representations to group algebra elements

Products in group algebra

  • Multiplication rules based on group operation, linear in both arguments
  • Distributive property: (a+b)c=ac+bc(a + b)c = ac + bc and a(b+c)=ab+aca(b + c) = ab + ac
  • Associativity inherited from group operation (ab)c=a(bc)(ab)c = a(bc)
  • Computation steps:
    1. Expand linear combinations
    2. Apply group multiplication
    3. Collect like terms
  • Examples: (1+g)(1−g)=1−g+g−g2(1 + g)(1 - g) = 1 - g + g - g^2 in cyclic group algebra, (s+r)(s−r)=s2−r2(s + r)(s - r) = s^2 - r^2 in dihedral group algebra

Key Terms to Review (17)

Basis of a Group Algebra: The basis of a group algebra is a set of elements that spans the entire group algebra space, allowing every element of the algebra to be expressed as a linear combination of these basis elements. This concept connects group theory and linear algebra, providing a way to represent group elements through matrices and facilitating the study of representations of groups in terms of vector spaces.
Character: In representation theory, a character is a function that assigns to each group element the trace of its corresponding matrix representation. Characters provide deep insights into the structure of representations, revealing information about their irreducibility and symmetry properties.
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 of a representation: The dimension of a representation refers to the size of the vector space associated with a representation of a group. It indicates the number of basis elements needed to describe the representation, and is crucial in understanding how the group acts on the space. This dimension connects to matrix representations, where the dimension corresponds to the number of rows or columns in the matrices, and plays a significant role in character theory by influencing the characters’ values.
Emil Artin: Emil Artin was a prominent mathematician known for his contributions to algebra, number theory, and representation theory. His work laid the foundation for understanding how abstract algebraic structures can be represented through matrices and functions, influencing various areas like group algebras, irreducible representations, regular representations, and L-functions in number theory.
Group Action: A group action is a formal way of describing how a group interacts with a set by associating each group element with a transformation of that set. This concept is crucial as it helps in understanding how symmetries can be represented and analyzed, linking group theory to geometry and other mathematical structures. Group actions lead to important ideas like orbits and stabilizers, allowing us to study the structure of groups in more depth.
Group Algebra: A group algebra is a mathematical structure formed from a group and a field, where elements of the group are treated as basis elements of a vector space over the field. This construction allows for the manipulation and analysis of group representations, leading to significant results in representation theory.
Homomorphism: A homomorphism is a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces, that respects the operations defined on those structures. This concept is crucial for understanding how different algebraic entities relate to each other, especially when exploring their properties and behaviors under transformations.
Induction: Induction is a method of reasoning that establishes the truth of a statement by proving it for a base case and then showing that if it holds for an arbitrary case, it also holds for the next case. This technique is especially useful in areas like representation theory, where it helps in constructing representations and understanding their properties, connecting foundational concepts with complex applications in group theory and algebra.
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.
Maschke's Theorem: Maschke's Theorem states that if a finite group is acting on a finite-dimensional vector space over a field whose characteristic does not divide the order of the group, then every representation of the group can be decomposed into a direct sum of irreducible representations. This theorem is fundamental in understanding the structure of representations, as it guarantees that every representation can be analyzed and simplified into simpler components, which is crucial for studying linear representations, matrix representations, and group algebras.
Number Theory: Number theory is a branch of pure mathematics that focuses on the properties and relationships of numbers, particularly integers. It encompasses various concepts such as divisibility, prime numbers, and congruences, which are fundamental in understanding more complex mathematical structures. Number theory often intersects with algebra and has applications in cryptography and coding theory, making it essential for mathematical research and practical applications.
Quantum Mechanics: Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the scale of atoms and subatomic particles. This theory introduces concepts such as wave-particle duality, uncertainty principle, and quantization, which are pivotal in understanding the behavior of particles and their interactions. Its principles have far-reaching implications across various fields, including representation theory, where it intersects with topics like matrix representations and orthogonality relations.
Schur's Lemma: Schur's Lemma is a fundamental result in representation theory that characterizes the homomorphisms between irreducible representations of a group or algebra. It states that if two irreducible representations are equivalent, then any intertwining operator between them is either an isomorphism or zero, providing crucial insights into the structure of representations and their relationships.
Semisimple Representation: A semisimple representation is a linear representation of a group that can be decomposed into a direct sum of irreducible representations. This means that every vector in the representation can be expressed as a sum of vectors from irreducible subspaces, which are invariant under the action of the group. Semisimple representations are significant because they allow for the simplification of complex representations and relate closely to the structure of the group algebra, enabling deeper understanding and analysis of group actions.
Unitary Representation: A unitary representation is a way of representing a group as a set of linear transformations on a complex inner product space, such that the group elements preserve the inner product. This concept is significant because it provides a framework for studying groups through their actions on vector spaces, connecting directly to matrix representations, tensor products, and the structure of weights and root systems in representation theory.
William Burnside: William Burnside was a British mathematician known for his contributions to group theory, particularly in representation theory. He is best known for Burnside's lemma, which provides a method for counting distinct objects under group actions, linking combinatorial aspects with the study of groups and their representations.
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