8.1 Basics of Group Representation Theory

Group representation 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 Schur's Lemma, 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 homomorphism $\rho$ from $G$ to $GL(V)$, the general linear group of invertible linear maps on a vector space $V$ over a field $F$ ($\mathbb{C}$ or $\mathbb{R}$)
• The dimension of the representation equals the dimension of the vector space $V$
• A subrepresentation of a representation $(\rho, V)$ is a subspace $W$ of $V$ that remains invariant under the group action, meaning $\rho(g)(W) \subseteq W$ for all $g \in G$
• A representation is irreducible when it has no proper, non-zero subrepresentations

Operations on group representations

• The direct sum of two representations $(\rho₁, V₁)$ and $(\rho₂, V₂)$ is the representation $(\rho₁ \oplus \rho₂, V₁ \oplus V₂)$ defined by $(\rho₁ \oplus \rho₂)(g)(v₁, v₂) = (\rho₁(g)(v₁), \rho₂(g)(v₂))$
  • Example: If $\rho₁$ is a representation of $G$ on $\mathbb{R}^2$ and $\rho₂$ is a representation of $G$ on $\mathbb{R}^3$, then $\rho₁ \oplus \rho₂$ is a representation of $G$ on $\mathbb{R}^5$
• Two representations $(\rho₁, V₁)$ and $(\rho₂, V₂)$ are equivalent if there exists an invertible linear map $\phi: V₁ \to V₂$ such that $\phi(\rho₁(g)(v)) = \rho₂(g)(\phi(v))$ for all $g \in G$ and $v \in V₁$
  • Example: The trivial representation and the sign representation of $S_3$ 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 $\rho: G \to GL(\mathbb{C})$ given by $\rho(g) = 1$ for all $g \in G$
• The regular representation of a finite group $G$ is the permutation representation $(\rho, \mathbb{C}[G])$ defined by $\rho(g)(h) = gh$ for all $g, h \in G$
  • Example: For $G = S_3$, 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 character of a representation $(\rho, V)$ is the function $\chi: G \to \mathbb{C}$ given by $\chi(g) = tr(\rho(g))$, where $tr$ denotes the trace of a linear map
• The character of a representation determines the representation up to equivalence
• The character table 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 $S_3$ 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

• Maschke's Theorem 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 $S_3$ over $\mathbb{C}$ is a direct sum of the trivial, sign, and standard representations

Schur's Lemma and irreducibility

• Schur's Lemma asserts that if $(\rho₁, V₁)$ and $(\rho₂, V₂)$ are irreducible representations of $G$ and $\phi: V₁ \to V₂$ is a $G$-equivariant linear map (meaning $\phi(\rho₁(g)(v)) = \rho₂(g)(\phi(v))$ for all $g \in G$ and $v \in V₁$), then either $\phi$ is an isomorphism or $\phi = 0$
  • Corollary: The only $G$-equivariant linear maps from an irreducible representation 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 $G = S_3$, the regular representation decomposes as the trivial representation $\oplus$ the sign representation $\oplus$ 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 $\mathbb{C}^{\times}$)
• Irreducible representations of $S_n$: The irreducible representations of the symmetric group $S_n$ are in bijection with the partitions of $n$, and their dimensions are given by the hook length formula
  • Example: $S_4$ 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 $D_{2n}$ 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 $(\rho₁, V₁)$ and $(\rho₂, V₂)$ are representations of $G$, their tensor product is the representation $(\rho₁ \otimes \rho₂, V₁ \otimes V₂)$ defined by $(\rho₁ \otimes \rho₂)(g)(v₁ \otimes v₂) = \rho₁(g)(v₁) \otimes \rho₂(g)(v₂)$
• 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 $|n-m|+1, |n-m|+3, \ldots, n+m-1$