8.2 Characters and the Weyl integral formula

Characters are the secret sauce of representation theory. They're like a fingerprint for each representation, telling us everything we need to know without getting bogged down in details. By studying characters, we can classify and compare representations easily.

The Weyl integral formula is a powerful tool for calculating characters of irreducible representations. It lets us express characters in terms of integrals over the maximal torus, simplifying computations and revealing deep connections between representation theory and Lie group structure.

Representation Character Properties

Definition and Trace Formula

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• The character of a representation $\rho: G \to GL(V)$ is a function $\chi_{\rho}: G \to \mathbb{C}$ defined by $\chi_{\rho}(g) = tr(\rho(g))$, where $tr$ denotes the trace of a linear operator
  • The trace of a linear operator is the sum of the diagonal entries of its matrix representation
  • Example: For a representation $\rho: G \to GL_2(\mathbb{C})$ with $\rho(g) = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the character value at $g$ is $\chi_{\rho}(g) = a + d$

Invariance Properties

• Characters are class functions, meaning they are constant on conjugacy classes of the group $G$
  • Conjugacy classes are sets of elements that are conjugate to each other, i.e., $\{hgh^{-1} | h \in G\}$ for some $g \in G$
  • Example: In the symmetric group $S_3$, the conjugacy classes are $\{e\}$, $\{(12), (13), (23)\}$, and $\{(123), (132)\}$. The character takes the same value on elements within each class
• Characters are invariant under equivalence of representations
  • Two representations $\rho_1$ and $\rho_2$ are equivalent if there exists an invertible linear map $A: V_1 \to V_2$ such that $A\rho_1(g) = \rho_2(g)A$ for all $g \in G$
  • If two representations are equivalent, their characters are equal: $\chi_{\rho_1} = \chi_{\rho_2}$

Operations on Characters

• The character of a direct sum of representations is the sum of the characters of the individual representations
  • For representations $\rho_1: G \to GL(V_1)$ and $\rho_2: G \to GL(V_2)$, the direct sum representation $\rho_1 \oplus \rho_2: G \to GL(V_1 \oplus V_2)$ has character $\chi_{\rho_1 \oplus \rho_2} = \chi_{\rho_1} + \chi_{\rho_2}$
  • Example: If $\chi_{\rho_1}(g) = a$ and $\chi_{\rho_2}(g) = b$, then $\chi_{\rho_1 \oplus \rho_2}(g) = a + b$
• The character of a tensor product of representations is the product of the characters of the individual representations
  • For representations $\rho_1: G \to GL(V_1)$ and $\rho_2: G \to GL(V_2)$, the tensor product representation $\rho_1 \otimes \rho_2: G \to GL(V_1 \otimes V_2)$ has character $\chi_{\rho_1 \otimes \rho_2} = \chi_{\rho_1} \cdot \chi_{\rho_2}$
  • Example: If $\chi_{\rho_1}(g) = a$ and $\chi_{\rho_2}(g) = b$, then $\chi_{\rho_1 \otimes \rho_2}(g) = ab$

Weyl Integral Formula for Characters

Weyl Integral Formula

• The Weyl integral formula expresses the character of an irreducible representation in terms of an integral over the maximal torus $T$ of the compact Lie group $G$
• For an irreducible representation $\rho_{\lambda}$ with highest weight $\lambda$, the Weyl integral formula states: $\chi_{\lambda}(\exp(X)) = \frac{\sum_{w \in W} \varepsilon(w)e^{i\langle w(\lambda+\rho),X\rangle}}{\sum_{w \in W} \varepsilon(w)e^{i\langle w(\rho),X\rangle}}$
  • $W$ is the Weyl group
  • $\varepsilon(w)$ is the sign of the Weyl group element $w$
  • $\rho$ is the Weyl vector (half the sum of positive roots)
  • $X \in \mathfrak{t}$ (the Lie algebra of $T$)

Weyl Denominator

• The denominator in the Weyl integral formula is the Weyl denominator, which can be expressed as a product over positive roots: $\prod_{\alpha \in \Delta^+} (e^{i\alpha(X)/2} - e^{-i\alpha(X)/2})$
• The Weyl denominator is independent of the specific irreducible representation and depends only on the root system of the Lie algebra
• Example: For the Lie group $SU(2)$, the Weyl denominator is $e^{iX/2} - e^{-iX/2}$, where $X \in \mathfrak{t} \cong \mathbb{R}$

Computational Advantage

• The Weyl integral formula allows for the computation of characters of irreducible representations without explicitly constructing the representation
• By expressing the character as an integral over the maximal torus, the formula reduces the complexity of the computation
• Example: For the Lie group $SU(3)$, the Weyl integral formula can be used to compute the characters of its irreducible representations, which are labeled by two non-negative integers $(m,n)$

Characters for Irreducible Representations

Equivalence and Distinctness

• Characters can be used to determine whether two representations are equivalent or distinct
• If two irreducible representations have the same character, they are equivalent
  • Equivalent representations have the same character values for all group elements
  • Example: For the Lie group $SU(2)$, the irreducible representations with highest weights $\lambda = m$ and $\lambda = -m$ have the same character and are equivalent
• Distinct irreducible representations have different characters
  • There exists at least one group element for which the character values differ
  • Example: For the Lie group $SU(3)$, the irreducible representations with highest weights $(m,n)$ and $(n,m)$ have different characters for $m \neq n$ and are distinct

Dimension Formula

• The dimension of an irreducible representation $\rho$ can be computed using the character: $\dim(\rho) = \chi_{\rho}(e)$, where $e$ is the identity element of the group
• The character value at the identity element gives the trace of the identity matrix, which equals the dimension of the representation space
• Example: For the standard representation of $SU(2)$ on $\mathbb{C}^2$, the character value at the identity is $\chi(e) = 2$, which is the dimension of the representation space

Orthogonality Relations

• Orthogonality relations for characters state that for irreducible characters $\chi_i$ and $\chi_j$, the integral $\int_G \chi_i(g)\chi_j(g)^* dg$ is equal to $1$ if $i = j$ and $0$ otherwise
  • $^*$ denotes complex conjugation
  • $dg$ is the normalized Haar measure on $G$
• The orthogonality relations express the idea that distinct irreducible characters are orthogonal with respect to the inner product on the space of class functions
• Example: For the Lie group $SU(2)$, the orthogonality relations can be used to show that the characters of the irreducible representations with highest weights $\lambda = m$ and $\lambda = n$ are orthogonal for $m \neq n$

Number of Irreducible Representations

• The number of irreducible representations of a compact Lie group is equal to the number of conjugacy classes
• Each irreducible representation corresponds to a unique conjugacy class, and vice versa
• Example: The Lie group $SU(2)$ has infinitely many conjugacy classes, corresponding to the infinitely many irreducible representations labeled by non-negative integers $\lambda = 0, 1, 2, \ldots$

Significance of Characters in Representation Theory

Classification and Study of Representations

• Characters provide a powerful tool for studying and classifying representations of compact Lie groups without explicitly constructing the representations
• The character table of a compact Lie group encodes essential information about its irreducible representations, including their dimensions and relations
  • The character table is a square matrix whose rows and columns are labeled by conjugacy classes and irreducible representations, respectively
  • The entries of the character table are the character values of each irreducible representation on each conjugacy class
• Example: The character table of the symmetric group $S_3$ completely describes its irreducible representations and their properties

Decomposition of Representations

• Characters can be used to decompose a given representation into irreducible components using the orthogonality relations and the inner product of characters
• The multiplicity of an irreducible representation $\rho_i$ in a representation $\rho$ can be computed as $\langle \chi_{\rho}, \chi_i \rangle = \int_G \chi_{\rho}(g)\chi_i(g)^* dg$
  • The multiplicity is the number of times the irreducible representation appears in the decomposition of $\rho$
  • If the multiplicity is zero, the irreducible representation does not appear in the decomposition
• Example: For the regular representation of a finite group $G$, which acts on the space of functions $f: G \to \mathbb{C}$, the decomposition into irreducible representations can be obtained using the character inner product

Weyl Character Formula

• The Weyl character formula, a generalization of the Weyl integral formula, expresses the character of an irreducible representation as a ratio of two Weyl-alternating sums
• The formula highlights the role of the Weyl group and root system in the representation theory of compact Lie groups
• For an irreducible representation with highest weight $\lambda$, the Weyl character formula states: $\chi_{\lambda}(e^X) = \frac{\sum_{w \in W} \varepsilon(w)e^{w(\lambda + \rho)(X)}}{\sum_{w \in W} \varepsilon(w)e^{w(\rho)(X)}}$
  • $\rho$ is the Weyl vector (half the sum of positive roots)
  • $X \in \mathfrak{t}$ (the Lie algebra of the maximal torus $T$)
• Example: For the Lie group $SU(2)$, the Weyl character formula gives the characters of the irreducible representations with highest weights $\lambda = m$ as $\chi_m(e^{iX}) = \frac{\sin((m+1)X)}{\sin(X)}$

Applications in Physics and Mathematics

• The study of characters is central to the application of representation theory in various fields, such as quantum mechanics and harmonic analysis on homogeneous spaces
• In quantum mechanics, irreducible representations of Lie groups are used to describe the symmetries of physical systems, and characters play a crucial role in determining the energy levels and transition probabilities
• In harmonic analysis, characters are used to study the decomposition of functions on homogeneous spaces (e.g., spheres, projective spaces) into irreducible components, which leads to important results in Fourier analysis and special functions
• Example: The characters of the irreducible representations of the rotation group $SO(3)$ are used to describe the angular momentum states in quantum mechanics and to study the spherical harmonics on the 2-sphere