🧩Representation Theory Unit 3 – Irreducible Representations

Key Concepts and Definitions

• Representation theory studies abstract algebraic structures by representing their elements as linear transformations of vector spaces
• Irreducible representation cannot be further decomposed into smaller representations
• Character of a representation is the trace of the corresponding matrix representation
• Schur's lemma states that any matrix that commutes with all matrices of an irreducible representation is a scalar multiple of the identity matrix
• Tensor product of representations allows for the construction of new representations from existing ones
• Direct sum of representations decomposes a representation into a sum of smaller representations
• Induced representation constructs a representation of a group from a representation of one of its subgroups
• Restricted representation obtains a representation of a subgroup from a representation of the larger group

Group Theory Foundations

• Group is a set equipped with a binary operation satisfying closure, associativity, identity, and inverse properties
• Abelian group has a commutative binary operation $(ab = ba)$
• Subgroup is a subset of a group that forms a group under the same binary operation
  • Lagrange's theorem states that the order of a subgroup divides the order of the group
• Coset of a subgroup $H$ in a group $G$ is the set of elements obtained by multiplying all elements of $H$ by a fixed element of $G$
• Normal subgroup is a subgroup $N$ such that $gNg^{-1} = N$ for all $g$ in the group
  • Quotient group $G/N$ can be formed when $N$ is a normal subgroup
• Conjugacy class consists of all elements that are conjugate to each other $(gxg^{-1})$
• Class function is a function that is constant on conjugacy classes

Constructing Irreducible Representations

• Maschke's theorem guarantees that every representation of a finite group is a direct sum of irreducible representations
• Regular representation is obtained by the action of the group on itself by left multiplication
  • Decomposes into a direct sum of all irreducible representations, each appearing with a multiplicity equal to its dimension
• Irreducible representations can be constructed using the method of induced representations
  • Start with a representation of a subgroup and induce it to the whole group
• Tensor product of irreducible representations can yield new irreducible representations
• Schur-Weyl duality relates the irreducible representations of the symmetric group and the general linear group
• Young tableaux provide a combinatorial method to construct and classify irreducible representations of the symmetric group
• Lie groups and Lie algebras have a correspondence between their irreducible representations

Properties of Irreducible Representations

• Schur's lemma implies that irreducible representations have no non-trivial invariant subspaces
  • Consequently, any matrix that commutes with all matrices of an irreducible representation is a scalar multiple of the identity
• Orthogonality relations state that matrix elements of irreducible representations are orthogonal
  • $\sum_g \overline{D_{ij}^{(\alpha)}(g)} D_{kl}^{(\beta)}(g) = \frac{|G|}{d_\alpha} \delta_{\alpha\beta} \delta_{ik} \delta_{jl}$
• Characters of irreducible representations are orthogonal
  • $\sum_g \overline{\chi^{(\alpha)}(g)} \chi^{(\beta)}(g) = |G| \delta_{\alpha\beta}$
• Irreducible characters uniquely determine the representation up to isomorphism
• Number of irreducible representations equals the number of conjugacy classes
• Dimension of an irreducible representation divides the order of the group

Character Tables and Their Uses

• Character table summarizes the characters of all irreducible representations of a group
  • Rows correspond to irreducible representations, columns to conjugacy classes
• Characters are class functions, constant on conjugacy classes
• Orthogonality relations of characters can be used to decompose a representation into irreducible components
• Regular character (character of the regular representation) equals the order of the group for the identity element and zero elsewhere
• Product of characters corresponds to the character of the tensor product representation
• Sum of squares of the dimensions of irreducible representations equals the order of the group
• Character tables can be used to determine the reducibility of a representation and the number of times each irreducible representation appears in its decomposition

Applications in Physics and Chemistry

• Symmetry groups play a crucial role in understanding physical systems
  • Crystallographic groups describe the symmetries of crystal lattices
  • Point groups classify the symmetries of molecules
• Irreducible representations of symmetry groups determine the allowed energy levels and transitions in quantum systems
  • Selection rules for electronic, vibrational, and rotational transitions can be derived from the irreducible representations
• Molecular orbitals transform according to irreducible representations of the molecular point group
  • Symmetry-adapted linear combinations (SALCs) of atomic orbitals form basis functions for the irreducible representations
• Jahn-Teller effect, the distortion of a non-linear molecule to remove degeneracy, can be explained using representation theory
• Irreducible tensor operators simplify the calculation of matrix elements in quantum mechanics
• Wigner-Eckart theorem relates matrix elements of tensor operators to reduced matrix elements and Clebsch-Gordan coefficients

Computational Techniques

• Character tables can be computed using the Burnside matrix or the Dixon-Schneider algorithm
• Irreducible representations can be constructed using the method of projectors
  • Projection operators $P^{(\alpha)}_{ij} = \frac{d_\alpha}{|G|} \sum_g \overline{D_{ij}^{(\alpha)}(g)} g$ project onto the irreducible subspaces
• Clebsch-Gordan coefficients, which describe the coupling of angular momenta, can be computed using the Racah formula or the Wigner 3j-symbols
• Wigner D-matrices, representing the elements of the rotation group SO(3), can be efficiently computed using recursive algorithms
• Fast Fourier transforms on finite groups can be used to compute convolutions and perform efficient matrix-vector multiplications
• Schur transforms, which block-diagonalize matrices according to irreducible representations, can be computed using the Wedderburn decomposition
• Computational group theory packages, such as GAP and Magma, provide efficient algorithms for working with representations and characters

Advanced Topics and Extensions

• Representation theory of infinite groups, such as Lie groups and algebraic groups, requires techniques from functional analysis and algebraic geometry
  • Unitary representations on Hilbert spaces are of particular importance in mathematical physics
• Modular representation theory studies representations over fields of positive characteristic
  • Brauer characters, defined as traces of matrices over splitting fields, play a role analogous to ordinary characters
• Representations of algebras, such as group algebras and Lie algebras, generalize the theory of group representations
  • Quiver representations, associated with directed graphs, have applications in algebraic geometry and quantum field theory
• Categorification lifts representation-theoretic concepts to a higher categorical level
  • Grothendieck groups of certain monoidal categories recover classical representation rings
• Geometric representation theory studies representations through their connections with algebraic geometry and symplectic geometry
  • Moment maps and symplectic reductions provide a geometric perspective on the construction of representations
• Representation stability investigates the asymptotic behavior of sequences of representations
  • Stable ranges and representation stability phenomena have been observed in various contexts, such as the cohomology of configuration spaces
• Categorified and quantum versions of representation theory, such as Khovanov homology and quantum groups, have led to new invariants and structures in low-dimensional topology and mathematical physics