Glossary

3.1 Definition and properties of irreducible representations

2 min read•july 25, 2024

Irreducible representations are the building blocks of group theory. They can't be broken down further and have unique properties like and . These representations are crucial for understanding group structure.

Examples range from the trivial representation to in quantum mechanics. For , all irreducible representations are one-dimensional. The number of irreducible representations links to the 's center, offering deep insights into group structure.

Irreducible Representations: Definition and Properties

Properties of irreducible representations

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  • cannot be decomposed into smaller representations, lacks proper non-trivial subrepresentations
  • Key properties demonstrate simplicity with no invariant subspaces except {0}\{0\} and the whole space
  • Schur's lemma states any intertwining operator is a scalar multiple of the identity
  • Orthogonality relations show matrix elements of distinct irreducible representations are orthogonal
  • can be finite or infinite-dimensional (Hilbert spaces)
  • Finite-dimensional irreducible representations are completely reducible into direct sums

Examples of irreducible representations

  • Trivial representation maps all group elements to 1, always irreducible for any group
  • generally reducible for non-trivial groups, decomposes into of all irreducible representations
  • [SU(2)](https://www.fiveableKeyTerm:su(2))[SU(2)](https://www.fiveableKeyTerm:su(2)) spin representations illustrate irreducibility in quantum mechanics
  • Representations of CnC_n demonstrate irreducibility in finite groups
  • Reducible representations include direct sum of non-isomorphic irreducible representations and tensor product of two irreducible representations

One-dimensionality in abelian groups

  • Proof outline demonstrates irreducible representations of abelian groups are one-dimensional:
    1. Consider irreducible representation ρ\rho of abelian group GG
    2. Show ρ(g)\rho(g) commutes with all ρ(h)\rho(h), h∈Gh \in G
    3. Apply Schur's lemma: ρ(g)\rho(g) must be scalar multiple of identity
    4. Conclude representation space is one-dimensional
  • Consequences reveal number of irreducible representations equals order of group for finite abelian groups
  • of irreducible representations form from GG to C∗\mathbb{C}^*

Irreducible representations vs group algebra centers

  • Group algebra C[G]\mathbb{C}[G] forms with basis elements corresponding to group elements
  • contains elements commuting with all elements in C[G]\mathbb{C}[G], spanned by sums over conjugacy classes
  • Irreducible representations correspond to in C[G]\mathbb{C}[G]
  • Center acts as scalar multiplication on irreducible representations
  • Number of irreducible representations equals dimension of the center
  • form basis for center of C[G]\mathbb{C}[G]
  • Characters of irreducible representations create orthonormal basis for class functions

Key Terms to Review (17)

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.
Center of Group Algebra: The center of a group algebra is the set of elements in the algebra that commute with every element of the algebra. In the context of irreducible representations, the center plays a crucial role in understanding how representations can be decomposed and analyzed, since elements from the center act as scalars on irreducible representations, preserving their structure and properties.
Characters: In representation theory, characters are complex-valued functions that provide a powerful way to study representations of groups by associating each group element with a trace of the corresponding linear transformation. Characters help to simplify the analysis of representations by enabling us to focus on their essential features, such as irreducibility and equivalence. They play a crucial role in connecting group theory to linear algebra, especially when examining properties like Maschke's theorem, irreducible representations, and geometric interpretations of representations.
Class Functions: Class functions are functions defined on the elements of a group that only depend on the conjugacy classes of those elements. They play a significant role in representation theory, particularly in analyzing representations and their properties, such as irreducibility and character theory. Class functions are particularly useful for studying symmetries and can be applied to derive important results like Burnside's theorem, facilitate understanding irreducible representations, and leverage Frobenius reciprocity in the context of group actions.
Cyclic Groups: A cyclic group is a type of group that can be generated by a single element, meaning every element in the group can be expressed as a power of that generator. These groups play a crucial role in various areas, such as understanding irreducible representations, character theory, and the applications of Frobenius reciprocity. The structure of cyclic groups helps to simplify complex problems by leveraging their inherent properties.
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.
Direct Sum: The direct sum is a construction in linear algebra that allows for the combination of two or more vector spaces into a new vector space, where each element is formed from a unique combination of elements from the component spaces. This concept plays a crucial role in understanding how representations can be decomposed into irreducible components, showcasing how different representations can coexist independently while contributing to a larger representation.
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.
Homomorphisms: Homomorphisms are structure-preserving maps between algebraic structures, such as groups, rings, or vector spaces, that respect the operations defined on these structures. They play a crucial role in understanding how different representations relate to each other and help establish connections between various algebraic concepts, especially in the context of irreducible representations, where they can show how these representations behave under certain transformations.
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.
Minimal left ideals: Minimal left ideals are the smallest non-zero left ideals within a given algebra, meaning they cannot be properly contained within any other left ideal. These ideals are essential in understanding irreducible representations, as they correspond to the simplest forms of representations that cannot be decomposed further. Recognizing minimal left ideals helps in identifying the building blocks of representations, leading to a better grasp of their properties and structure.
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.
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.
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.
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.
Su(2): su(2) is a special unitary group that consists of 2x2 complex matrices with a trace of zero and unit determinant. It is fundamental in the representation theory of groups and plays a crucial role in quantum mechanics, particularly in describing spin-1/2 particles. Understanding su(2) is essential for exploring irreducible representations and calculating Clebsch-Gordan coefficients.
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.
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