⭕Groups and Geometries Unit 12 – Finite Group Representation Theory

Key Concepts and Definitions

• Group consists of a set $G$ together with a binary operation $*$ satisfying closure, associativity, identity, and inverse properties
• Representation of a group $G$ is a homomorphism $\rho: G \rightarrow GL(V)$ where $V$ is a vector space and $GL(V)$ is the general linear group of invertible linear transformations on $V$
  • Homomorphism preserves the group structure, i.e., $\rho(g_1 * g_2) = \rho(g_1) \cdot \rho(g_2)$ for all $g_1, g_2 \in G$
• Character of a representation $\rho$ is the function $\chi_\rho: G \rightarrow \mathbb{C}$ defined by $\chi_\rho(g) = \text{tr}(\rho(g))$, where $\text{tr}$ denotes the trace
• Irreducible representation cannot be decomposed into a direct sum of smaller representations
• Conjugacy class of an element $g \in G$ is the set $\{hgh^{-1} : h \in G\}$
• Class function is a function $f: G \rightarrow \mathbb{C}$ that is constant on conjugacy classes, i.e., $f(hgh^{-1}) = f(g)$ for all $g, h \in G$

Foundations of Group Theory

• Subgroup $H$ of a group $G$ is a subset that forms a group under the same operation as $G$
• Cosets of a subgroup $H$ in $G$ are sets of the form $gH = \{gh : h \in H\}$ for $g \in G$
  • Cosets partition the group $G$ into disjoint sets of equal size
• Normal subgroup $N$ of $G$ satisfies $gNg^{-1} = N$ for all $g \in G$
  • Quotient group $G/N$ can be formed by considering cosets of $N$ as elements
• Homomorphism $\varphi: G \rightarrow H$ is a function that preserves the group operation, i.e., $\varphi(g_1 * g_2) = \varphi(g_1) \cdot \varphi(g_2)$ for all $g_1, g_2 \in G$
  • Kernel of a homomorphism $\varphi$ is the set $\text{ker}(\varphi) = \{g \in G : \varphi(g) = e_H\}$, where $e_H$ is the identity element of $H$
• Isomorphism is a bijective homomorphism, and isomorphic groups have the same structure
• Direct product of groups $G$ and $H$ is the group $G \times H = \{(g, h) : g \in G, h \in H\}$ with component-wise operation

Introduction to Representation Theory

• Representation of a group $G$ on a vector space $V$ is a homomorphism $\rho: G \rightarrow GL(V)$
  • $\rho(g)$ is a linear transformation on $V$ for each $g \in G$
• Degree of a representation is the dimension of the vector space $V$
• Equivalent representations $\rho_1$ and $\rho_2$ satisfy $\rho_2(g) = A^{-1}\rho_1(g)A$ for some invertible matrix $A$ and all $g \in G$
• Trivial representation maps every group element to the identity transformation
• Regular representation is the permutation representation of $G$ acting on itself by left multiplication
• Subrepresentation of $\rho: G \rightarrow GL(V)$ is a representation $\rho': G \rightarrow GL(W)$ where $W$ is a subspace of $V$ invariant under $\rho(g)$ for all $g \in G$
• Direct sum of representations $\rho_1 \oplus \rho_2$ acts on the direct sum of the corresponding vector spaces

Characters and Character Tables

• Character of a representation $\rho$ is the function $\chi_\rho: G \rightarrow \mathbb{C}$ defined by $\chi_\rho(g) = \text{tr}(\rho(g))$
  • Characters are class functions, i.e., constant on conjugacy classes
• Character table of a group $G$ is a matrix whose rows correspond to irreducible characters and columns correspond to conjugacy classes
  • Entries are the values of the irreducible characters on representative elements of each conjugacy class
• Orthogonality relations for characters state that irreducible characters are orthonormal with respect to a specific inner product
  • $\langle \chi_i, \chi_j \rangle = \delta_{ij}$ for irreducible characters $\chi_i$ and $\chi_j$, where $\delta_{ij}$ is the Kronecker delta
• Number of irreducible representations equals the number of conjugacy classes
• Sum of squares of the degrees of irreducible representations equals the order of the group
• Regular character $\chi_{\text{reg}}$ satisfies $\chi_{\text{reg}}(e) = |G|$ and $\chi_{\text{reg}}(g) = 0$ for $g \neq e$

Irreducible Representations

• Irreducible representation cannot be decomposed into a direct sum of smaller representations
  • No non-trivial invariant subspaces under the action of the group
• Every representation can be written as a direct sum of irreducible representations
• Schur's Lemma states that any linear map between irreducible representations that commutes with the group action is either zero or an isomorphism
  • Consequence: the only matrices that commute with all matrices of an irreducible representation are scalar multiples of the identity
• Characters of irreducible representations are orthonormal with respect to the inner product $\langle f, g \rangle = \frac{1}{|G|} \sum_{x \in G} f(x)\overline{g(x)}$
• Irreducible characters form a basis for the space of class functions
• Number of times an irreducible representation appears in a representation is given by the inner product of their characters

Applications in Geometry

• Symmetry group of a geometric object (polygon, polyhedron, etc.) is the group of transformations that leave the object invariant
  • Representations of the symmetry group can be used to analyze the object's properties
• Character theory can be used to determine the number of fixed points of a group action
  • Burnside's Lemma: the number of orbits is the average number of fixed points over all group elements
• Representations can be used to construct symmetric and antisymmetric tensors
  • Symmetric tensors are invariant under permutations of indices, while antisymmetric tensors change sign under odd permutations
• Invariant theory studies polynomials that are invariant under a group action
  • Molien series is a generating function that encodes the dimensions of the spaces of invariant polynomials of each degree
• Representation theory can be used to analyze the vibrations and normal modes of molecules and crystals
  • Symmetry-adapted linear combinations of atomic orbitals form basis functions for irreducible representations

Problem-Solving Techniques

• Determine the conjugacy classes and the size of each class
  • Elements in the same conjugacy class have the same character values
• Construct the character table by calculating the traces of representative matrices for each conjugacy class
  • Use orthogonality relations to check the table's consistency
• Decompose a representation into irreducible representations using the inner product of characters
  • Multiply the character of the representation by each irreducible character and divide by the group order
• Apply Schur's Lemma to determine the structure of commutant algebras and intertwining maps
  • Commutant algebra consists of matrices that commute with all matrices of the representation
• Use Molien's theorem to find the generating function for the dimensions of invariant polynomial spaces
  • Substitute the character values into the Molien series formula
• Utilize Burnside's Lemma to count the number of orbits under a group action
  • Calculate the average number of fixed points over all group elements

Advanced Topics and Extensions

• Projective representations are homomorphisms from a group to the projective general linear group $PGL(V)$
  • Useful for studying groups with nontrivial cohomology, such as the symmetric group $S_n$ for $n \geq 4$
• Unitary representations are representations where the matrices $\rho(g)$ are unitary, i.e., $\rho(g)^{-1} = \rho(g)^\dagger$
  • Unitary representations are important in quantum mechanics and harmonic analysis
• Lie groups are continuous groups with a smooth manifold structure
  • Representations of Lie groups are crucial in theoretical physics and harmonic analysis
• Induced representations are constructed by starting with a representation of a subgroup and extending it to the whole group
  • Frobenius reciprocity relates induced representations to restricted representations
• Modular representation theory studies representations over fields of positive characteristic
  • Decomposition of representations can be more complicated due to the presence of non-semisimple modules
• Categorification of representation theory replaces vector spaces with categories and linear maps with functors
  • Provides a deeper understanding of representation-theoretic concepts and connections to other areas of mathematics