Representation Theory

Group theory studies symmetry in the abstract. Linear algebra studies matrices acting on vectors. Representation Theory is the bridge between them: it studies how abstract groups can be represented as matrices. This allows us to use the powerful tools of linear algebra (trace, determinant, eigenvalues) to solve problems in abstract algebra and physics.

The Basic Idea

A collection of symmetries (a Group $G$) can often be represented by a set of linear transformations on a vector space $V$. A representation is a homomorphism $\rho: G \to GL(V)$. This means that for every group element $g$, there is an invertible matrix $\rho(g)$ such that the group composition is preserved: $\rho(g_1 g_2) = \rho(g_1) \rho(g_2)$

Irreducible Representations (Irreps)

Just as an integer can be broken down into prime factors, a representation can often be broken down into smaller, simpler representations. If a representation cannot be broken down further, it is called irreducible.

The Maschke Theorem states that for finite groups (over fields like $\mathbb{C}$), every representation is a direct sum of irreducible ones. This is effectively the “fundamental theorem of arithmetic” for representations.

Characters: Data Compression for Symmetries

Working with full matrices for every group element is computationally expensive. Character Theory simplifies this by focusing only on the trace of the matrices. The character of a representation $\rho$ is the function $\chi: G \to \mathbb{C}$ defined by: $\chi(g) = \text{tr}(\rho(g))$

Characters are “class functions”—they are the same for elements in the same conjugacy class. This remarkably compact representation contains almost all the information about the representation.

import numpy as np

# Representation of the Cyclic Group C4 
# Generated by a rotation of 90 degrees
def get_c4_rep(k):
  # k is the power (0, 1, 2, 3)
  theta = k * (np.pi / 2)
  return np.array([[np.cos(theta), -np.sin(theta)],
                   [np.sin(theta),  np.cos(theta)]])

# Calculate characters for each element
characters = []
for k in range(4):
  rho_g = get_c4_rep(k)
  chi_g = np.trace(rho_g)
  characters.append(np.round(chi_g, 2))

print("Group Elements: [e, r, r^2, r^3]")
print("Characters:    ", characters)

# Note: The trace of the identity (e) is 2, 
# which is the dimension of the representation.

Symmetry in Physics and Chemistry

In quantum mechanics, particles are described by wavefunctions. If a physical system has a certain symmetry (like rotating a crystal), the wavefunction must transform according to a representation of that symmetry group.

• Molecular Vibrations: Predicting which vibrations are “infrared active” in a molecule is done by decomposing the representation of the molecule’s symmetry group.
• Particle Physics: The “Standard Model” is built on the representations of the groups $SU(3) \times SU(2) \times U(1)$. Particles are literally just vectors in the spaces where these groups act!

Exercises