8.2 Representation rings and character theory

Representation rings and character theory are crucial tools in understanding group actions and symmetries. They provide a powerful algebraic framework for analyzing the structure of representations, encoding how different representations combine and interact.

In the context of equivariant K-theory, representation rings serve as a foundation for studying more complex structures. They connect abstract algebraic concepts to geometric and topological properties, bridging the gap between group theory and the study of vector bundles over spaces with group actions.

Representation Rings of Lie Groups

Definition and Structure

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• The representation ring of a compact Lie group G, denoted R(G), is the Grothendieck ring of finite-dimensional complex representations of G
• Elements of R(G) are formal differences of isomorphism classes of representations, with addition given by direct sum and multiplication given by tensor product
• R(G) is a commutative ring with identity, where the identity element is the trivial representation (the representation that maps every group element to the identity matrix)
• The ring R(G) is a free abelian group generated by the irreducible representations of G
  • An irreducible representation is a representation that cannot be decomposed into a direct sum of smaller representations
  • Every representation can be written as a direct sum of irreducible representations
• The product of two irreducible representations can be decomposed into a direct sum of irreducible representations using the Clebsch-Gordan coefficients
  • The Clebsch-Gordan coefficients are the multiplicities of irreducible representations in the tensor product of two irreducible representations
  • Example: In SU(2), the tensor product of two spin-1/2 representations decomposes into a direct sum of a spin-0 and a spin-1 representation

Significance and Applications

• The representation ring captures essential information about the representation theory of G, such as the dimensionality and characters of representations
• R(G) encodes the algebraic structure of the representations, including how they combine under direct sum and tensor product
• The representation ring is a powerful tool for studying the symmetries and invariants of physical systems with G-symmetry
  • Example: In quantum mechanics, the representation ring of the symmetry group of a system determines the possible energy levels and transition rules
• R(G) is also connected to other areas of mathematics, such as algebraic geometry and combinatorics, through its relation to characters and symmetric functions

Classical Lie Group Representations

Unitary Groups

• The representation ring of the unitary group U(n) is isomorphic to the ring of symmetric polynomials in n variables, R(U(n)) ≅ Z[x₁, ..., xₙ]^Sₙ
  • Symmetric polynomials are polynomials that are invariant under permutations of the variables
  • The isomorphism is given by mapping a representation to its character, which is a symmetric polynomial
• The irreducible representations of U(n) are indexed by Young diagrams or partitions of integers
  • A Young diagram is a collection of boxes arranged in left-justified rows, with each row having at most as many boxes as the row above it
  • A partition of an integer n is a sequence of positive integers (λ₁, ..., λ₊) such that λ₁ ≥ ... ≥ λ₊ and λ₁ + ... + λ₊ = n
  • Example: The partition (3, 2, 1) corresponds to the Young diagram ☐☐☐ ☐☐ ☐
• The character of an irreducible representation corresponding to a partition λ is given by the Schur polynomial sλ(x₁, ..., xₙ)
  • Schur polynomials are a special basis for the ring of symmetric polynomials, with many nice algebraic and combinatorial properties
  • Example: The Schur polynomial s₍₂,₁₎(x₁, x₂, x₃) = x₁²x₂ + x₁²x₃ + x₁x₂² + x₂²x₃ + x₁x₃² + x₂x₃²

Orthogonal and Symplectic Groups

• The representation ring of the orthogonal group O(n) is a subring of R(U(n)), generated by exterior powers of the standard representation
  • The standard representation of O(n) is the n-dimensional representation given by matrix multiplication on vectors in R^n
  • The exterior powers Λ^k(R^n) are representations of O(n) corresponding to k-dimensional subspaces of R^n
  • R(O(n)) ≅ Z[λ₁, ..., λ₊]/(λ₊^2 - 1) for n = 2k+1, and R(O(n)) ≅ Z[λ₁, ..., λ₊]/(λ₊^2 - 2λ₊) for n = 2k, where λᵢ is the i-th exterior power of the standard representation
• The representation ring of the symplectic group Sp(n) is also a subring of R(U(n)), generated by symmetric powers of the standard representation
  • The standard representation of Sp(n) is the 2n-dimensional representation given by matrix multiplication on vectors in C^2n, preserving a symplectic form
  • The symmetric powers Sym^k(C^2n) are representations of Sp(n) corresponding to k-fold symmetric tensor products of C^2n
  • R(Sp(n)) ≅ Z[σ₁, ..., σₙ], where σᵢ is the i-th symmetric power of the standard representation
• The representations of O(n) and Sp(n) can be studied using similar techniques as for U(n), such as character theory and combinatorics of Young diagrams
  • Example: The irreducible representations of O(n) are indexed by partitions λ with λᵢ = 0 for i > ⌊n/2⌋, and the irreducible representations of Sp(n) are indexed by partitions λ with λᵢ ≤ n

Character Theory for Representations

Definition and Properties

• Characters are class functions on the group G, constant on conjugacy classes, and uniquely determine representations up to isomorphism
  • A class function is a function f: G → C that is constant on each conjugacy class of G
  • Two representations are isomorphic if and only if they have the same character
• The character of a representation ρ: G → GL(V) is the function χ_ρ: G → C given by χ_ρ(g) = Tr(ρ(g)), where Tr denotes the trace
  • The trace of a matrix is the sum of its diagonal entries
  • Example: For the standard representation of U(n), the character χ_std(g) is the sum of the eigenvalues of the matrix g
• Characters are additive under direct sum and multiplicative under tensor product of representations
  • For representations ρ₁ and ρ₂, χ_{ρ₁ ⊕ ρ₂} = χ_ρ₁ + χ_ρ₂ and χ_{ρ₁ ⊗ ρ₂} = χ_ρ₁ · χ_ρ₂
• The irreducible characters form an orthonormal basis for the space of class functions with respect to the inner product ⟨χ, ψ⟩ = (1/|G|) ∑_{g∈G} χ(g)ψ(g)*
  • The inner product of two class functions is a measure of their overlap or similarity
  • The irreducible characters are orthogonal and normalized with respect to this inner product, making them a convenient basis for studying class functions

Applications and Character Tables

• The multiplicity of an irreducible representation in a given representation can be computed using the inner product of their characters
  • For a representation ρ and an irreducible representation π, the multiplicity of π in ρ is given by ⟨χ_ρ, χ_π⟩
  • This allows for the decomposition of a representation into its irreducible components
• Character tables, which list the values of irreducible characters on conjugacy classes, can be used to deduce various properties of representations, such as dimensionality, irreducibility, and decomposition into irreducible components
  • The character table of a group G is a square matrix whose rows are indexed by the irreducible characters and whose columns are indexed by the conjugacy classes
  • The (i,j)-entry of the character table is the value of the i-th irreducible character on the j-th conjugacy class
  • Example: The character table of the symmetric group S₃ is | e | (12) | (123) | --|-----|------|-------| 1 | 1 | 1 | 1 | ε | 1 | -1 | 1 | ρ | 2 | 0 | -1 | where e is the identity, (12) is a transposition, and (123) is a 3-cycle, and 1, ε, and ρ are the trivial, sign, and 2-dimensional irreducible representations, respectively
• Character theory provides a powerful tool for studying representations of finite groups and compact Lie groups, with applications in physics, chemistry, and combinatorics
  • Example: In quantum chemistry, character theory is used to determine the symmetry properties of molecular orbitals and to predict selection rules for electronic transitions

Representation Rings vs Equivariant K-theory

Equivariant K-theory

• Equivariant K-theory, denoted K_G(X), is a generalization of topological K-theory that incorporates the action of a compact Lie group G on a space X
  • Topological K-theory, K(X), is a cohomology theory that studies vector bundles over a space X
  • A G-equivariant vector bundle over X is a vector bundle π: E → X together with an action of G on E that is compatible with the action of G on X and the projection π
• Elements of K_G(X) are formal differences of isomorphism classes of G-equivariant vector bundles over X, with addition given by direct sum and multiplication given by tensor product
  • Two G-equivariant vector bundles are isomorphic if there is a G-equivariant isomorphism between them
  • The direct sum and tensor product of G-equivariant vector bundles are again G-equivariant vector bundles
• K_G(X) is a ring, with the zero element given by the trivial vector bundle and the identity element given by the rank-1 trivial vector bundle
  • The trivial vector bundle over X is the product bundle X × C^n, with G acting trivially on the fibers
• Equivariant K-theory satisfies many of the same properties as ordinary K-theory, such as homotopy invariance, Bott periodicity, and the existence of Chern characters
  • Example: If X is a point, then K_G(pt) is isomorphic to the representation ring R(G), as G-equivariant vector bundles over a point are just representations of G

Relation to Representation Rings

• The equivariant Chern character, ch_G: K_G(X) → H_G^*(X; Q), is a ring homomorphism from equivariant K-theory to equivariant cohomology with rational coefficients
  • Equivariant cohomology, H_G^*(X), is a cohomology theory that incorporates the action of G on X
  • The equivariant Chern character is a generalization of the ordinary Chern character, which maps K(X) to the ordinary cohomology H^*(X; Q)
• For a point space X = pt, K_G(pt) is isomorphic to the representation ring R(G), and the equivariant Chern character reduces to the ordinary character map ch: R(G) → H^*(BG; Q), where BG is the classifying space of G
  • The classifying space BG is a topological space that classifies principal G-bundles, and its cohomology H^*(BG) encodes information about the representation theory of G
  • The character map sends a representation to its character, viewed as an element of H^*(BG; Q) via the Chern character
• The equivariant Chern character allows for the computation of equivariant K-theory in terms of equivariant cohomology, which is often more tractable
  • Many tools from algebraic topology, such as spectral sequences and localization theorems, can be applied to compute equivariant cohomology
  • The equivariant Chern character provides a link between these computations and the study of equivariant vector bundles and representations
• The Atiyah-Segal completion theorem relates the representation ring R(G) to the equivariant K-theory of the classifying space BG, stating that K_G^*(BG) is isomorphic to the completion of R(G) with respect to the augmentation ideal
  • The augmentation ideal of R(G) is the kernel of the dimension homomorphism dim: R(G) → Z, which sends a representation to its dimension
  • The completion of R(G) with respect to the augmentation ideal is a topological ring that captures additional structure beyond the representation ring itself
  • Example: For G = U(1), the representation ring R(U(1)) is isomorphic to the ring of Laurent polynomials Z[t, t^(-1)], and its completion with respect to the augmentation ideal is the ring of formal power series Z[[t]]