🧩Representation Theory Unit 2 – Linear Representations of Groups

Key Concepts and Definitions

• Linear representation maps group elements to invertible matrices while preserving the group operation
• Character of a representation $\chi(g)$ equals the trace of the matrix representing group element $g$
• Irreducible representation cannot be decomposed into smaller representations
  • Fundamental building blocks for constructing all representations of a group
• Schur's Lemma states that any linear map between irreducible representations is either zero or an isomorphism
• Maschke's Theorem guarantees that every representation of a finite group can be decomposed into irreducible representations
• Regular representation $\rho_G$ acts on the group algebra $\mathbb{C}[G]$ by right multiplication
• Tensor product $V \otimes W$ of two representations creates a new representation acting on the tensor product space
• Direct sum $V \oplus W$ of two representations creates a new representation acting on the direct sum space

Group Theory Foundations

• Group $G$ consists of a set with a binary operation satisfying closure, associativity, identity, and inverse properties
• Abelian groups (commutative) satisfy $gh = hg$ for all $g, h \in G$
• Subgroup $H \leq G$ is a subset closed under the group operation and inverse
  • Lagrange's Theorem states that the order of a subgroup divides the order of the group
• Cosets $gH = \{gh : h \in H\}$ partition the group into disjoint sets of equal size
• Normal subgroup $N \trianglelefteq G$ satisfies $gNg^{-1} = N$ for all $g \in G$
  • Quotient group $G/N$ formed by cosets of a normal subgroup is itself a group
• Homomorphism $\varphi: G \to H$ preserves the group operation, i.e., $\varphi(g_1g_2) = \varphi(g_1)\varphi(g_2)$
• Isomorphism is a bijective homomorphism, denoted $G \cong H$

Vector Spaces and Linear Algebra Review

• Vector space $V$ over field $\mathbb{F}$ satisfies closure under addition and scalar multiplication
• Subspace $W \leq V$ is a subset closed under vector addition and scalar multiplication
• Linear independence means a set of vectors has no non-trivial linear combination equal to zero
• Basis is a linearly independent spanning set for a vector space
  • Dimension $\dim(V)$ equals the number of vectors in any basis
• Linear map (transformation) $T: V \to W$ preserves vector addition and scalar multiplication
  • Kernel $\ker(T) = \{v \in V : T(v) = 0\}$ and image $\text{im}(T) = \{T(v) : v \in V\}$ are subspaces
• Matrix representation $[T]_\mathcal{B}^\mathcal{C}$ of a linear map depends on the choice of bases $\mathcal{B}$ and $\mathcal{C}$
• Eigenvalues $\lambda$ and eigenvectors $v$ satisfy $Av = \lambda v$ for a square matrix $A$

Linear Representations: Introduction

• Linear representation $(V, \rho)$ of a group $G$ consists of a vector space $V$ and a homomorphism $\rho: G \to \text{GL}(V)$
  • $\rho(g)$ is an invertible linear map on $V$ for each $g \in G$
  • Homomorphism property: $\rho(g_1g_2) = \rho(g_1)\rho(g_2)$ for all $g_1, g_2 \in G$
• Degree of a representation is the dimension of the vector space $V$
• Faithful representation $\rho$ is injective, i.e., $\rho(g) = I$ if and only if $g = e$
• Equivalent representations $(V, \rho)$ and $(W, \psi)$ satisfy $\psi(g) = T\rho(g)T^{-1}$ for some invertible map $T: V \to W$
  • Character tables can distinguish non-equivalent representations
• Trivial representation maps every group element to the identity matrix
• Permutation representation acts on a set $X$ by permuting its elements according to the group action

Character Theory

• Character $\chi_\rho(g) = \text{tr}(\rho(g))$ is the trace of the matrix representing $g$ in representation $\rho$
  • Characters are class functions, constant on conjugacy classes
• Irreducible characters form an orthonormal basis for the space of class functions under the inner product $\langle \chi, \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \chi(g)\overline{\psi(g)}$
• Character table organizes characters of irreducible representations as columns and conjugacy classes as rows
  • Number of irreducible representations equals the number of conjugacy classes
• Orthogonality relations: $\sum_{g \in G} \chi_i(g)\overline{\chi_j(g)} = |G|\delta_{ij}$ and $\sum_i \chi_i(g)\overline{\chi_i(h)} = |C_G(g)|\delta_{gh}$
• Regular character $\chi_\text{reg}(e) = |G|$ and $\chi_\text{reg}(g) = 0$ for $g \neq e$
• Induced character $\chi_\rho^G(g) = \frac{1}{|H|} \sum_{x \in G, xgx^{-1} \in H} \chi_\rho(xgx^{-1})$ lifts a character from a subgroup $H$ to $G$

Irreducible Representations

• Irreducible representation (irrep) $V$ has no non-trivial subrepresentations
  • Every representation decomposes into a direct sum of irreps
• Schur's Lemma implies that any intertwining map (morphism of representations) between irreps is either zero or an isomorphism
  • Corollary: irreps are determined up to isomorphism by their characters
• Regular representation $(\mathbb{C}[G], \rho_\text{reg})$ decomposes as $\mathbb{C}[G] \cong \bigoplus_i V_i^{\oplus \dim V_i}$, summing over all irreps $V_i$
• Tensor product $V \otimes W$ of irreps may decompose into a direct sum of irreps
  • Clebsch-Gordan coefficients express this decomposition explicitly
• Restriction $\text{Res}_H^G(\rho)$ of a $G$-representation to a subgroup $H$ is a representation of $H$
• Induced representation $\text{Ind}_H^G(\rho)$ lifts an $H$-representation to a $G$-representation
  • Frobenius reciprocity relates restriction and induction: $\text{Hom}_G(\text{Ind}_H^G(V), W) \cong \text{Hom}_H(V, \text{Res}_H^G(W))$

Applications and Examples

• Dihedral group $D_{2n}$ of symmetries of a regular $n$-gon has $n/2 + 3$ conjugacy classes (for even $n$)
  • Irreps are the trivial, sign, and 2-dimensional rotation representations
• Quaternion group $Q_8 = \{\pm 1, \pm i, \pm j, \pm k\}$ has 5 conjugacy classes and 5 irreps (4 linear and 1 of degree 2)
• Symmetric group $S_n$ acts on $\mathbb{C}^n$ by permuting coordinates (permutation representation)
  • Irreps are indexed by integer partitions of $n$ (Young diagrams)
• Unitary group $U(n)$ acts on $\mathbb{C}^n$ preserving the Hermitian inner product
  • Irreps correspond to symmetric tensor powers of the standard representation
• Compact Lie groups (matrix groups) have finite-dimensional irreps
  • Representation theory extends to infinite dimensions for other Lie groups
• Fourier analysis on finite abelian groups $G$ decomposes functions $f: G \to \mathbb{C}$ using characters $\chi$
  • Fourier transform $\hat{f}(\chi) = \sum_{g \in G} f(g)\overline{\chi(g)}$ and inversion formula $f(g) = \frac{1}{|G|} \sum_\chi \hat{f}(\chi)\chi(g)$

Advanced Topics and Extensions

• Projective representations are homomorphisms into the projective linear group $\text{PGL}(V)$
  • Cohomology theory classifies projective representations up to equivalence
• Modular representation theory studies representations over fields of positive characteristic
  • Decomposition into irreps may fail, but indecomposable representations exist
• Unitary representations on Hilbert spaces are important in quantum mechanics and harmonic analysis
  • Stone-von Neumann Theorem classifies irreps of the Heisenberg group
• Representations of algebras (e.g., Lie algebras, Hopf algebras) generalize group representations
  • Representations of $\mathfrak{sl}_2(\mathbb{C})$ are crucial in physics and combinatorics
• Category-theoretic viewpoint: representation theory as the study of functors from groups (or algebras) to vector spaces
  • Deligne's Theorem on tensor categories classifies certain representation categories
• Langlands program relates representation theory of Lie groups to automorphic forms and number theory
  • Reciprocity laws and functoriality conjectures connect different areas of mathematics
• Geometric representation theory studies actions of groups on algebraic varieties
  • Borel-Weil Theorem realizes irreps as sections of line bundles over flag varieties