6.3 Finite-dimensional representations and their characters

Finite-dimensional representations are key to understanding semisimple Lie algebras. They're classified by their highest weights, which are elements of the Cartan subalgebra's dual space. This classification helps us study the structure of these representations.

Characters are functions that encode essential info about representations, like dimension and weight space decomposition. The Weyl character formula is a powerful tool for computing characters of irreducible representations, connecting representation theory to group theory and topology.

Irreducible Representations of Lie Algebras

Classification of Finite-Dimensional Irreducible Representations

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• Semisimple Lie algebras are direct sums of simple Lie algebras, and their finite-dimensional representations are completely reducible
• Finite-dimensional irreducible representations of semisimple Lie algebras are uniquely determined by their highest weight, an element of the dual space of the Cartan subalgebra
  • The highest weight is the weight of the highest weight vector, which is annihilated by all positive root vectors
  • Example: In the case of $\mathfrak{sl}(3, \mathbb{C})$, the fundamental weights are $\omega_1 = (1, 0)$ and $\omega_2 = (0, 1)$ in the basis of simple roots
• The fundamental weights form a basis for the space of integral weights and play a crucial role in the classification of irreducible representations
  • Integral weights are weights whose pairing with any coroot is an integer
  • Example: For $\mathfrak{sl}(3, \mathbb{C})$, the irreducible representation with highest weight $\lambda = m\omega_1 + n\omega_2$ is denoted by $V(m, n)$

Weyl Group Action and Dimension Formula

• The Weyl group acts on the space of weights, and the dominant integral weights (those in the fundamental Weyl chamber) parametrize the finite-dimensional irreducible representations
  • The fundamental Weyl chamber is a cone in the weight space defined by the simple roots
  • Example: For $\mathfrak{sl}(3, \mathbb{C})$, the fundamental Weyl chamber is defined by $\lambda_1 \geq \lambda_2 \geq 0$
• The dimension of an irreducible representation with highest weight $\lambda$ is given by the Weyl dimension formula, which involves the positive roots of the Lie algebra
  • The Weyl dimension formula is $\dim V(\lambda) = \prod_{\alpha \in \Delta^+} \frac{\langle \lambda + \rho, \alpha \rangle}{\langle \rho, \alpha \rangle}$, where $\rho$ is half the sum of positive roots
  • Example: For $\mathfrak{sl}(3, \mathbb{C})$, the dimension of $V(m, n)$ is $\frac{1}{2}(m+1)(n+1)(m+n+2)$

Characters of Representations

Weyl Character Formula

• The character of a representation is a function on the Lie algebra that encodes essential information about the representation, such as its dimension and weight space decomposition
• The Weyl character formula expresses the character of an irreducible representation with highest weight $\lambda$ as a ratio of two alternating sums over the Weyl group
  • The formula is $\chi_\lambda(h) = \frac{\sum_{w \in W} (-1)^{\ell(w)} e^{w(\lambda + \rho)(h)}}{\sum_{w \in W} (-1)^{\ell(w)} e^{w\rho(h)}}$, where $h$ is an element of the Cartan subalgebra and $\ell(w)$ is the length of the Weyl group element $w$
  • Example: For $\mathfrak{sl}(2, \mathbb{C})$, the character of the irreducible representation $V(n)$ is $\chi_n(h) = \frac{e^{(n+1)h} - e^{-(n+1)h}}{e^h - e^{-h}}$

Properties and Applications of Characters

• The numerator of the Weyl character formula involves the exponential of the highest weight $\lambda$ and the Weyl group action, while the denominator involves the positive roots of the Lie algebra
• The Weyl character formula can be used to compute the character of any finite-dimensional representation by expressing it as a sum of characters of irreducible representations
  • Example: The character of the adjoint representation of $\mathfrak{sl}(3, \mathbb{C})$ is $\chi_{\text{ad}}(h) = 2 + e^{\alpha_1(h)} + e^{\alpha_2(h)} + e^{-\alpha_1(h)} + e^{-\alpha_2(h)} + e^{(\alpha_1 + \alpha_2)(h)} + e^{-(\alpha_1 + \alpha_2)(h)}$
• The characters of representations are class functions, meaning they are constant on conjugacy classes of the Lie group associated with the Lie algebra
  • This property allows characters to be used to study the representation theory of Lie groups
  • Example: The characters of representations of $\mathfrak{sl}(2, \mathbb{C})$ are determined by their values on the diagonal matrices $\text{diag}(t, -t)$, which form a maximal torus in $SL(2, \mathbb{C})$

Characters and Representation Rings

Representation Rings and Character Maps

• The representation ring of a Lie algebra is a ring generated by the isomorphism classes of finite-dimensional representations, with addition given by direct sum and multiplication given by tensor product
  • The representation ring encodes the algebraic structure of representations
  • Example: In the representation ring of $\mathfrak{sl}(2, \mathbb{C})$, $[V(1)] \otimes [V(1)] = [V(2)] \oplus [V(0)]$, where $[V(n)]$ denotes the isomorphism class of the irreducible representation with highest weight $n$
• Characters of representations are ring homomorphisms from the representation ring to the ring of class functions on the Lie group
  • The ring of class functions is the set of functions on the Lie group that are constant on conjugacy classes, with pointwise addition and multiplication
  • Example: The character of the tensor product of two representations is the product of their characters, i.e., $\chi_{V \otimes W}(g) = \chi_V(g) \chi_W(g)$ for any group element $g$

Character Tables and Representation Theory

• The character map, which sends a representation to its character, is an injective ring homomorphism from the representation ring to the ring of class functions
  • This allows the study of representations through their characters
  • Example: The character map for $\mathfrak{sl}(2, \mathbb{C})$ sends $[V(n)]$ to the character $\chi_n(t) = \frac{t^{n+1} - t^{-(n+1)}}{t - t^{-1}}$, where $t$ is a diagonal matrix $\text{diag}(t, t^{-1})$
• The character table of a Lie algebra is a square matrix whose rows are indexed by the irreducible representations and whose columns are indexed by the conjugacy classes of the Lie group, with entries given by the character values
  • The character table encodes the structure of the representation ring and provides a way to determine the decomposition of representations into irreducible components
  • Example: The character table of $\mathfrak{sl}(2, \mathbb{C})$ has entries $\chi_n(t^m) = \frac{t^{m(n+1)} - t^{-m(n+1)}}{t^m - t^{-m}}$, where $n$ indexes the irreducible representations and $m$ indexes the conjugacy classes

Character Theory Applications

Decomposition of Representations

• Character theory is a powerful tool for studying the decomposition of representations into irreducible components, as the character of a representation determines it up to isomorphism
• The multiplicity of an irreducible representation in a given representation can be computed using the inner product of characters, which involves integrating the product of their characters over the Lie group
  • The multiplicity formula is $m_\lambda = \frac{1}{|W|} \int_T \overline{\chi_\lambda(t)} \chi_V(t) dt$, where $T$ is a maximal torus and $|W|$ is the order of the Weyl group
  • Example: The multiplicity of $V(n)$ in the tensor product $V(m) \otimes V(k)$ for $\mathfrak{sl}(2, \mathbb{C})$ is given by the Clebsch-Gordan coefficient $\langle m, k | n \rangle$

Tensor Products and Branching Rules

• The tensor product of two irreducible representations can be decomposed into a direct sum of irreducible representations using the Clebsch-Gordan coefficients, which can be determined from the character table
  • The Clebsch-Gordan coefficients are the multiplicities of irreducible representations in the tensor product
  • Example: For $\mathfrak{sl}(2, \mathbb{C})$, the tensor product $V(m) \otimes V(n)$ decomposes as $\bigoplus_{k=|m-n|}^{m+n} V(k)$, with multiplicities given by the Clebsch-Gordan coefficients
• The branching rules for restricting representations from a Lie algebra to a subalgebra can be determined using character theory and the Weyl character formula
  • Branching rules describe how irreducible representations of the larger algebra decompose into irreducible representations of the subalgebra
  • Example: The branching rule for restricting representations of $\mathfrak{sl}(3, \mathbb{C})$ to $\mathfrak{sl}(2, \mathbb{C})$ embedded as the upper-left block is given by $V(m, n) \downarrow \mathfrak{sl}(2, \mathbb{C}) = \bigoplus_{k=0}^n V(m+k)$

Invariant Subspaces and Homogeneous Spaces

• Character theory can be used to study the invariant subspaces and quotient spaces of representations, as well as to compute the Euler characteristic of homogeneous spaces associated with the Lie group
  • Invariant subspaces are subspaces of a representation that are preserved by the action of the Lie algebra or Lie group
  • Example: The symmetric and antisymmetric tensors are invariant subspaces of the tensor product of the standard representation of $\mathfrak{sl}(n, \mathbb{C})$ with itself
• Homogeneous spaces are quotients of Lie groups by closed subgroups, and their Euler characteristic can be computed using the Weyl character formula and the Lefschetz fixed-point theorem
  • The Euler characteristic is a topological invariant that measures the "size" of a space
  • Example: The Grassmannian $Gr(k, n)$ of $k$-dimensional subspaces in $\mathbb{C}^n$ is a homogeneous space for $SL(n, \mathbb{C})$, and its Euler characteristic is the binomial coefficient $\binom{n}{k}$