Group theory is a powerful tool in algebraic combinatorics. It helps us understand group actions on sets by studying their linear transformations on vector spaces. This approach reveals hidden symmetries and structures in combinatorial objects.
Representations allow us to break down complex group actions into simpler, irreducible parts. By analyzing characters and using key theorems like Maschke's and , we can classify and understand these fundamental building blocks of group actions.
Group representations and their properties
Definition and key properties of group representations
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A representation of a group G is a ρ from G to GL(V), the general linear group of invertible linear maps on a vector space V over a field F (C or R)
The dimension of the representation equals the dimension of the vector space V
A subrepresentation of a representation (ρ,V) is a subspace W of V that remains invariant under the group action, meaning ρ(g)(W)⊆W for all g∈G
A representation is irreducible when it has no proper, non-zero subrepresentations
Operations on group representations
The of two representations (ρ1,V1) and (ρ2,V2) is the representation (ρ1⊕ρ2,V1⊕V2) defined by (ρ1⊕ρ2)(g)(v1,v2)=(ρ1(g)(v1),ρ2(g)(v2))
Example: If ρ1 is a representation of G on R2 and ρ2 is a representation of G on R3, then ρ1⊕ρ2 is a representation of G on R5
Two representations (ρ1,V1) and (ρ2,V2) are equivalent if there exists an invertible linear map ϕ:V1→V2 such that ϕ(ρ1(g)(v))=ρ2(g)(ϕ(v)) for all g∈G and v∈V1
Example: The trivial representation and the sign representation of S3 are not equivalent, even though they have the same dimension
Constructing and analyzing group representations
Constructing basic representations
The trivial representation of a group G is the one-dimensional representation ρ:G→GL(C) given by ρ(g)=1 for all g∈G
The regular representation of a finite group G is the permutation representation (ρ,C[G]) defined by ρ(g)(h)=gh for all g,h∈G
Example: For G=S3, the regular representation has dimension 6 and decomposes into the trivial representation, the sign representation, and two copies of the standard representation
Analyzing representations using characters
The of a representation (ρ,V) is the function χ:G→C given by χ(g)=tr(ρ(g)), where tr denotes the trace of a linear map
The character of a representation determines the representation up to equivalence
The of a finite group is the table whose rows are indexed by the conjugacy classes of G and whose columns are the irreducible characters of G
Example: The character table of S3 has 3 rows (identity, transpositions, 3-cycles) and 3 columns (trivial, sign, standard representations)
Fundamental theorems in representation theory
Maschke's Theorem and complete reducibility
states that if G is a finite group and F is a field whose characteristic does not divide ∣G∣, then every representation of G over F is completely reducible (a direct sum of irreducible representations)
Example: Every representation of S3 over C is a direct sum of the trivial, sign, and standard representations
Schur's Lemma and irreducibility
Schur's Lemma asserts that if (ρ1,V1) and (ρ2,V2) are irreducible representations of G and ϕ:V1→V2 is a G-equivariant linear map (meaning ϕ(ρ1(g)(v))=ρ2(g)(ϕ(v)) for all g∈G and v∈V1), then either ϕ is an or ϕ=0
Corollary: The only G-equivariant linear maps from an to itself are scalar multiples of the identity
Counting irreducible representations
The number of irreducible representations of a finite group G (up to equivalence) equals the number of conjugacy classes of G
The sum of the squares of the dimensions of the irreducible representations of a finite group G equals ∣G∣
The regular representation of a finite group G is isomorphic to the direct sum of all irreducible representations of G, each occurring with multiplicity equal to its dimension
Example: For G=S3, the regular representation decomposes as the trivial representation ⊕ the sign representation ⊕ two copies of the standard representation
Classifying irreducible representations
Irreducible representations of specific groups
Irreducible representations of abelian groups: Every irreducible representation of a finite abelian group is one-dimensional, and they are given by the group's characters (homomorphisms from the group to C×)
Irreducible representations of Sn: The irreducible representations of the Sn are in bijection with the partitions of n, and their dimensions are given by the hook length formula
Example: S4 has 5 irreducible representations corresponding to the partitions [4], [3, 1], [2, 2], [2, 1, 1], and [1, 1, 1, 1]
Irreducible representations of dihedral groups: The dihedral group D2n has 2 one-dimensional irreducible representations and (n−1) two-dimensional irreducible representations if n is odd, or (n/2−1) two-dimensional irreducible representations and 2 one-dimensional irreducible representations if n is even
Tensor products and Clebsch-Gordan coefficients
Tensor products of representations: If (ρ1,V1) and (ρ2,V2) are representations of G, their tensor product is the representation (ρ1⊗ρ2,V1⊗V2) defined by (ρ1⊗ρ2)(g)(v1⊗v2)=ρ1(g)(v1)⊗ρ2(g)(v2)
The irreducible representations occurring in the tensor product can be determined using the Clebsch-Gordan coefficients
Example: For SU(2), the tensor product of the irreducible representations with dimensions n and m decomposes into irreducible representations with dimensions ∣n−m∣+1,∣n−m∣+3,…,n+m−1
Key Terms to Review (18)
Burnside: Burnside refers to Burnside's lemma, a key result in group theory that provides a way to count distinct objects under group actions. It states that the number of distinct orbits of a set acted on by a group is equal to the average number of points fixed by the elements of the group, giving an effective tool for counting in combinatorics.
Character: In mathematics, particularly in representation theory, a character is a homomorphism from a group to the multiplicative group of complex numbers. This means that characters provide a way to study the structure of a group through its representations, allowing for an understanding of how group elements act on vector spaces. Characters are crucial for analyzing representations because they encapsulate important information about the group's structure and its symmetries.
Character Table: A character table is a mathematical tool used in representation theory to summarize the characters of all irreducible representations of a finite group. It provides important information about the group's structure, including the dimensions of its representations and how they relate to each other, especially in the context of symmetric groups and their properties.
Cyclic group: 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 powers (or multiples) of that generator. This structure is fundamental in various areas of mathematics, as cyclic groups serve as the simplest types of groups, forming the building blocks for more complex group structures and having important applications in combinatorics, representation theory, and counting problems.
Direct Sum: The direct sum is a way to combine two or more mathematical structures, such as vector spaces or groups, into a new structure that maintains the properties of the original ones. In the context of representation theory, the direct sum of representations allows for the construction of a new representation that captures the behavior of each individual representation simultaneously, creating a richer overall picture.
Finite-dimensional representation: A finite-dimensional representation is a way to express a group as linear transformations on a finite-dimensional vector space. This concept is essential in understanding how groups act on vector spaces, as it provides a bridge between abstract algebra and linear algebra. By studying these representations, one can uncover properties of the group through matrix representations and gain insights into its structure and behavior.
Frobenius: Frobenius refers to a key concept in algebra and combinatorics related to the representation of symmetric functions and group representations. Named after the mathematician Ferdinand Frobenius, it is often associated with the Frobenius characteristic map that connects symmetric functions to representations of symmetric groups, and plays an important role in understanding the structure of representations and characters in group theory.
Group Algebra: A group algebra is a mathematical structure that combines elements of a group with coefficients from a field, allowing for the construction of linear combinations of group elements. This concept connects algebraic structures with representation theory, enabling the study of group actions in a more manageable way through linear algebra. Group algebras are essential in understanding characters and representations of finite groups, and they play a significant role in the development of Hopf algebras as well.
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. In the context of group theory, it means that the operation in the first group corresponds to the operation in the second group when applying the homomorphism. This concept is crucial because it helps us understand how different algebraic structures relate to each other and enables the transfer of properties and results between them.
Irreducible Representation: An irreducible representation of a group is a homomorphism from the group to the general linear group of a vector space that has no proper subrepresentation. This means that the only invariant subspaces under the action of the group are trivial (the zero space and the whole space), making it a building block for understanding how groups can act on vector spaces. Irreducible representations play a crucial role in character theory, allow for the classification of representations, and are essential in analyzing complex representations through simpler ones.
Isomorphism: Isomorphism refers to a structural similarity between two mathematical objects, indicating that they can be transformed into each other through a bijective mapping that preserves the relevant operations and relations. This concept is crucial in understanding the equivalence of different structures, revealing that while they may appear different, their underlying properties are essentially the same.
Linear transformation: A linear transformation is a mathematical function between two vector spaces that preserves the operations of vector addition and scalar multiplication. This means that if you apply the transformation to a combination of vectors, it will yield the same result as transforming each vector separately and then combining the results. Linear transformations play a crucial role in understanding how groups can act on vector spaces and are fundamental in representation theory.
Maschke's Theorem: Maschke's Theorem states that if a finite group acts on a finite-dimensional vector space over a field whose characteristic does not divide the order of the group, then the representation is completely reducible. This means that any representation can be decomposed into a direct sum of irreducible representations, ensuring that every representation can be simplified and analyzed through its building blocks. This theorem is crucial in understanding how groups can be represented in vector spaces, and it links group theory with linear algebra.
Module: A module is a mathematical structure that generalizes vector spaces by allowing scalars to come from a ring instead of a field. This structure captures the essence of linear algebra while extending it to contexts where the usual properties of fields may not apply, making modules essential in areas such as representation theory and algebraic geometry.
Representation: In mathematics, particularly in group theory, representation refers to a way to express elements of a group as linear transformations of a vector space. This allows us to study group properties through matrices and linear algebra, making complex group operations more manageable and understandable.
Schur's Lemma: Schur's Lemma is a fundamental result in representation theory that describes the behavior of homomorphisms between irreducible representations of a group. It states that if two irreducible representations are equivalent, any intertwining operator (homomorphism) between them is either zero or an isomorphism. This concept highlights the unique structure of irreducible representations and plays a crucial role in understanding how these representations interact.
Symmetric group: The symmetric group is a fundamental concept in abstract algebra that consists of all possible permutations of a finite set of elements, forming a group under the operation of composition. This group captures the notion of rearranging objects and plays a crucial role in combinatorics, representation theory, and many other areas of mathematics.
Symmetry in Combinatorics: Symmetry in combinatorics refers to the invariance of a mathematical object under certain transformations, such as rotations, reflections, or translations. This property is crucial because it allows mathematicians to simplify complex problems by recognizing that symmetric structures can lead to equivalent configurations, reducing the total number of distinct arrangements to consider.