Representation Theory

Representation Theory is the study of abstract algebraic structures by representing their elements as linear transformations of vector spaces. This technique effectively “linearizes” group theory, allowing us to use the powerful tools of matrix algebra and eigenvalues to solve problems in group theory, chemistry, and quantum mechanics.

Group Representations

A Representation of a group $G$ on a vector space $V$ over a field $k$ is a group homomorphism: $\rho :G\to GL(V)$ where $GL(V)$ is the general linear group of invertible linear transformations on $V$.

• The dimension of $V$ is called the dimension (or degree) of the representation.
• For each $g\in G$, $\rho (g)$ is an invertible matrix (if $V$ is finite dimensional).

G-Modules

An alternative way to view a representation is as a $G$-module. $V$ is a $G$-module if we can “multiply” vectors in $V$ by elements of $G$ such that the group action is linear. This allows us to apply the tools of module theory and homological algebra to groups.

Irreducible Representations (Irreps)

A representation is irreducible if the only subspaces of $V$ that are invariant under the action of all $\rho (g)$ are $\{0\}$ and $V$ itself.

• Maschke’s Theorem: If $G$ is a finite group and the characteristic of $k$ does not divide the order of $G$, then every representation of $G$ is a direct sum of irreducible representations. This is known as semi-simplicity.

Schur’s Lemma

Schur’s Lemma is a fundamental result about the morphisms between irreducible representations:

1. If ${\rho }_1$ and ${\rho }_2$ are irreducible representations and $f:V_1\to V_2$ is a $G$-linear map, then $f$ is either zero or an isomorphism.
2. If $k$ is algebraically closed, any $G$-linear map $f:V\to V$ from an irrep to itself is a scalar multiple of the identity: $f=\lambda I$.

Character Theory

The character of a representation $\rho$ is the function ${\chi }_{\rho }:G\to k$ defined by the trace: ${\chi }_{\rho }(g)=\text{Tr}(\rho (g))$ Characters are class functions, meaning they are constant on conjugacy classes ($\chi (hg{h}^{-1})=\chi (g)$).

Orthogonality Relations

The set of irreducible characters forms an orthonormal basis for the space of class functions under the inner product: $⟨\chi ,\psi ⟩=\frac{1}{|G|}{\sum }_{g\in G}\chi (g)\overline{\psi (g)}$ This allows us to decompose any representation into its irreducible components simply by computing traces.

The Group Algebra $k[G]$

The Group Algebra is the vector space with the elements of $G$ as a basis, where multiplication is extended linearly from the group operation.

• The study of representations of $G$ is equivalent to the study of modules over the ring $k[G]$.
• The number of distinct irreducible representations of a finite group $G$ is equal to the number of conjugacy classes of $G$.

Python: Computing a Group Character

We can represent group elements as matrices and compute their trace in Python.

import numpy as np

# A representation of the Cyclic Group C2 = {e, a}
# rho(e) = Identity, rho(a) = Refection matrix
rho_e = np.array([[1, 0], [0, 1]])
rho_a = np.array([[0, 1], [1, 0]])

def character(matrix):
    return np.trace(matrix)

print(f"Character of e: {character(rho_e)}")
print(f"Character of a: {character(rho_a)}")

# These values (2, 0) describe the character 'chi' of this representation.

Significance in Physics

In quantum mechanics, the state space of a system is a vector space, and the symmetries of the system (like rotations or translations) form a group. The physical observables must be invariant under these group actions, making Representation Theory the primary tool for classifying particles and understanding atomic spectra.

Exercise