🧩Representation Theory Unit 9 – Finite Group Representations

Key Concepts and Definitions

• Representation theory studies abstract algebraic structures by representing their elements as linear transformations of vector spaces
• A representation of a group $G$ is a homomorphism $\rho: G \to GL(V)$ from $G$ to the general linear group of a vector space $V$
• The dimension of the representation is the dimension of the vector space $V$
• A subrepresentation is a subspace of $V$ that is invariant under the action of $G$
• An irreducible representation has no non-trivial subrepresentations
• The character of a representation $\rho$ is the function $\chi_\rho: G \to \mathbb{C}$ given by $\chi_\rho(g) = \text{tr}(\rho(g))$, where $\text{tr}$ denotes the trace
• Schur's lemma states that any homomorphism between irreducible representations is either zero or an isomorphism

Group Theory Foundations

• Group theory is the study of algebraic structures called groups, which consist of a set and a binary operation satisfying certain axioms (associativity, identity, inverses)
• Examples of groups include the integers under addition $(\mathbb{Z}, +)$, the nonzero real numbers under multiplication $(\mathbb{R}^*, \times)$, and the symmetry group of a regular polygon
• A subgroup is a subset of a group that forms a group under the same operation
• Lagrange's theorem states that the order of a subgroup divides the order of the group
• Cosets are equivalence classes of a group with respect to a subgroup, and they partition the group
• Normal subgroups are subgroups invariant under conjugation, and they give rise to quotient groups
• The isomorphism theorems relate subgroups, quotient groups, and homomorphisms between groups

Introduction to Representations

• Representation theory provides a way to study abstract groups using linear algebra
• A matrix representation of a group $G$ is a homomorphism $\rho: G \to GL_n(\mathbb{C})$, where $GL_n(\mathbb{C})$ is the group of $n \times n$ invertible matrices over the complex numbers
• The regular representation is a representation of a group $G$ on the vector space $\mathbb{C}[G]$ of formal linear combinations of elements of $G$
• The permutation representation is a representation of a group $G$ acting on a set $X$, given by permutation matrices
• Representations can be constructed using the induced representation and the tensor product of representations
• Schur's orthogonality relations describe the orthogonality of matrix coefficients of irreducible representations

Character Theory

• The character of a representation $\rho$ is the function $\chi_\rho: G \to \mathbb{C}$ given by $\chi_\rho(g) = \text{tr}(\rho(g))$
• Characters are class functions, meaning they are constant on conjugacy classes
• The character table of a group encodes the values of the irreducible characters on each conjugacy class
• Orthogonality relations for characters state that irreducible characters are orthonormal with respect to a certain inner product
• The number of irreducible representations of a finite group equals the number of conjugacy classes
• Characters can be used to decompose a representation into irreducible components and to test for irreducibility
• The Frobenius-Schur indicator distinguishes between real, complex, and quaternionic representations

Irreducible Representations

• An irreducible representation (or simple module) has no non-trivial subrepresentations
• Every representation can be decomposed uniquely (up to isomorphism) as a direct sum of irreducible representations
• Schur's lemma implies that the endomorphism algebra of an irreducible representation is a division algebra over $\mathbb{C}$
• The group algebra $\mathbb{C}[G]$ decomposes as a direct sum of matrix algebras corresponding to the irreducible representations
• Irreducible characters are the trace functions of the irreducible representations
• The regular representation decomposes into a direct sum of all irreducible representations, with multiplicities equal to their dimensions
• Tensor products of irreducible representations can be decomposed using the Clebsch-Gordan coefficients

Applications in Physics and Chemistry

• Representation theory is used extensively in quantum mechanics to study symmetries of physical systems
• The symmetry group of a molecule determines its vibrational and rotational spectra
• Crystal structures are classified by their space groups, which are studied using representation theory
• The angular momentum operators in quantum mechanics form a representation of the rotation group $SO(3)$
• The Wigner-Eckart theorem relates matrix elements of tensor operators to Clebsch-Gordan coefficients
• Selection rules for atomic and molecular transitions are derived using representation theory
• The Pauli exclusion principle and the periodic table can be understood using the representation theory of the symmetric group

Computational Methods

• Character tables and representation matrices can be computed using computer algebra systems (GAP, Magma, Sage)
• The Dixon-Schneider algorithm computes the character table of a finite group from its multiplication table
• The Meat-axe algorithm decomposes a representation into irreducible components by finding invariant subspaces
• The Schreier-Sims algorithm computes a base and strong generating set for a permutation group, enabling efficient computation with subgroups and cosets
• Modular representation theory studies representations over fields of positive characteristic, which often requires computational methods
• Computational methods are used to study representations of large finite groups, such as the Monster group and the Baby Monster group

Advanced Topics and Open Problems

• Representation growth studies the asymptotic behavior of the number of irreducible representations of a group as a function of their dimension
• The representation zeta function encodes the representation growth of a group and is related to the abscissa of convergence
• The Kirillov orbit method relates irreducible representations of a Lie group to orbits of its coadjoint action on the dual of its Lie algebra
• The Langlands program is a far-reaching network of conjectures connecting representation theory to number theory and automorphic forms
• Kazhdan's property (T) is a representation-theoretic property of groups with applications in geometry, ergodic theory, and combinatorics
• The Baum-Connes conjecture relates the K-theory of the reduced C*-algebra of a group to its equivariant K-homology
• Open problems include the classification of irreducible representations of the symmetric groups, the existence of non-linear similarity-invariant norms on group algebras, and the asymptotic representation growth of arithmetic groups