2.1 Definition and examples of linear representations
2 min read•july 25, 2024
Linear representations map group elements to linear transformations, preserving group structure. This powerful tool allows us to study abstract groups using familiar linear algebra techniques, bridging the gap between group theory and linear algebra.
In practice, we verify if a function is a by checking key properties. These include the property, linearity of transformations, preservation of identity, and invertibility. These checks ensure the representation accurately captures the group's structure.
Fundamentals of Linear Representations
Linear representations of groups
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- Linear representation of a group maps group elements to linear transformations preserves group structure (homomorphism) (rotation groups, symmetry groups)
- Homomorphism from group G to GL(V) of V preserves algebraic structure
- Each group element corresponds to linear transformation on V maintains group properties
Group elements and linear transformations
- Group elements correspond to linear maps preserves vector space structure
- Identity element maps to identity transformation leaves vectors unchanged
- Group operation corresponds to composition of linear maps maintains group structure
- expresses linear transformations as matrices in GL(n, F) for n-dimensional vector space over field F
Vector spaces for group representations
- V acts as domain for group action via linear transformations
- Characteristics include (finite or infinite) and underlying field (real, complex, quaternions)
- Examples: or for finite-dimensional representations, function spaces for
- Basis choice affects matrix representation of group elements influences computations
Functions as linear group representations
- Trivial representation maps all group elements to identity transformation preserves structure trivially
- acts on group algebra by left multiplication captures group structure faithfully
- describes group elements as permutations on a set (symmetric groups)
- of symmetric group maps even permutations to 1, odd to -1 captures parity information
Verifying Linear Representations
Determine if a given function is a linear representation of a group
- Check homomorphism property for all g, h in G preserves group operation
- Confirm linearity of transformations for all g in G, v, w in V, and scalars a, b
- Verify preservation of identity where e is the identity element of G
- Check invertibility of for all g in G ensures bijective mapping
- Ensure dimension consistency all map V to itself preserving vector space structure
Key Terms to Review (19)
Artin: In representation theory, Artin refers to a class of representations that can be associated with a group or algebra, particularly in the context of linear representations. Artin's work has significantly influenced the understanding of how groups act on vector spaces, providing a framework for analyzing their structure through linear transformations.
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.
Faithful Representation: A faithful representation refers to a linear representation of a group that accurately reflects the structure of that group without any distortion. This means that different elements in the group are represented by distinct transformations, ensuring that the representation captures the essence of how the group operates. In a faithful representation, the only way for a transformation to act as the identity transformation is if the corresponding group element is also the identity element, establishing a clear one-to-one correspondence between group elements and their representations.
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.
Frobenius: Frobenius refers to a fundamental concept in representation theory that connects group representations and the way representations can be induced from a subgroup to a larger group. This principle not only establishes a method for constructing representations but also leads to deeper insights about the relationships between different groups and their actions. The Frobenius reciprocity theorem plays a crucial role in this framework, providing a connection between induced and restricted representations, making it essential for understanding linear representations and their applications.
General Linear Group: The general linear group, denoted as GL(n, F), is the group of all n x n invertible matrices with entries from a field F. This group plays a critical role in both algebra and representation theory, as it describes linear transformations that preserve vector spaces and enables the study of their symmetries and structures.
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.
Infinite-dimensional representations: Infinite-dimensional representations refer to representations of groups or algebras that act on vector spaces with infinite dimensions. These representations extend the concept of finite-dimensional representations, allowing for a richer structure that can model more complex systems. They are crucial in areas such as quantum mechanics and functional analysis, where infinite-dimensional spaces naturally arise.
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.
Linear representation: A linear representation is a way of expressing a group through linear transformations on a vector space, enabling the group structure to be analyzed via matrices and vector operations. This concept connects abstract group theory with concrete mathematical operations, allowing one to use techniques from linear algebra to study groups. Linear representations are particularly important because they facilitate understanding how groups can act on vector spaces, thereby revealing structural properties of both the groups and the spaces they act upon.
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.
Matrix Representation: Matrix representation refers to the way in which a group or algebra can be expressed as matrices acting on vector spaces. This concept allows abstract algebraic structures to be studied through linear transformations, linking group actions to linear algebra. It connects closely with notions like homomorphisms and isomorphisms, showcasing how these relationships preserve structure, and it forms the basis for understanding linear representations and character theory, which are crucial for analyzing the properties of groups through their representations.
Permutation representation: A permutation representation is a specific type of linear representation of a group where the group acts on a vector space by permuting its basis elements. This means that each group element corresponds to a linear transformation that rearranges the vectors in the space, effectively showcasing the group's structure through these actions. Understanding permutation representations is crucial for grasping how groups can be represented in a linear algebraic setting and has practical applications in finite group theory.
Reducible Representation: A reducible representation is a type of linear representation of a group in which the vector space can be expressed as a direct sum of two or more nontrivial invariant subspaces. This means that there exist proper subspaces of the representation that are preserved under the action of the group, allowing for a decomposition into simpler components. Understanding reducible representations is crucial when analyzing the structure and behavior of linear representations and their equivalence.
Regular Representation: The regular representation of a group is a specific type of linear representation where the group acts on itself by left multiplication. This construction allows one to view the group as a matrix representation, which is particularly useful for analyzing its structure and understanding its representations more generally.
Representation space: A representation space is a vector space associated with a group representation that captures how the elements of the group act on it through linear transformations. This concept is essential as it provides a framework for analyzing the structure of a group by studying its action on vector spaces, thus linking abstract group theory to concrete linear algebraic structures. The properties and behaviors of this space are crucial for understanding both linear representations and the characters of these representations.
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.
Sign Representation: Sign representation refers to a specific way in which a group (often a finite group) can be expressed as matrices acting on a vector space. This representation is significant because it allows the abstract algebraic structure of the group to be translated into a linear framework, making it easier to study group properties through linear algebra techniques. Understanding sign representations can illuminate how symmetries and transformations relate to mathematical objects.
Vector Space: A vector space is a mathematical structure formed by a collection of vectors, which can be added together and multiplied by scalars. This concept is foundational in linear algebra and underpins many areas in mathematics, including representation theory. The properties of vector spaces, such as closure under addition and scalar multiplication, are essential when discussing linear representations and irreducible representations, as they provide the necessary framework for manipulating and understanding these concepts.