Understanding Group Homomorphisms and Irreducible Representations

What We're Solving:

We need to create a function that maps each element of the symmetric group S₃ (the group of all permutations of 3 elements) to specific 2×2 matrices, preserving the group structure. This mapping will give us the "standard irreducible representation" of S₃.

The Approach:

Think of this as creating a "translation dictionary" between two different mathematical languages! We're taking abstract permutations and representing them as concrete matrices that behave the same way under multiplication. This is incredibly powerful because it lets us study group properties using linear algebra tools.

The key insight is that S₃ has a 2-dimensional irreducible representation that captures the "essential" symmetry of the group in matrix form.

Step-by-Step Solution:

Step 1: Understand what S₃ looks like S₃ has 6 elements: the identity, two 3-cycles, and three 2-cycles (transpositions). In cycle notation: e, (123), (132), (12), (13), (23).

Step 2: Find the standard irreducible representation matrices The 2-dimensional standard irreducible representation uses these specific 2×2 matrices:

• Identity → [[1,0],[0,1]]
• 3-cycles → rotation matrices involving ω = e^(2πi/3) (cube roots of unity)
• Transpositions → reflection-type matrices

Step 3: Set up SageMath code structure We'll create a dictionary mapping each S₃ element to its corresponding matrix, then define the homomorphism function.

Step 4: Implement the homomorphism ```python

Define S_3

S3 = SymmetricGroup(3)

Define the cube root of unity

omega = exp(2piI/3)

Create the matrix representations

rho_matrices = { S3(()): matrix([[1, 0], [0, 1]]), # identity S3((1,2,3)): matrix([[omega, 0], [0, omega^2]]), # (123) S3((1,3,2)): matrix([[omega^2, 0], [0, omega]]), # (132) S3((1,2)): matrix([[0, 1], [1, 0]]), # (12) S3((1,3)): matrix([[omega^2, 0], [0, omega]]) * matrix([[0, 1], [1, 0]]), # (13) S3((2,3)): matrix([[omega, 0], [0, omega^2]]) * matrix([[0, 1], [1, 0]]) # (23) }

Define the homomorphism

def rho(g): return rho_matrices[g] ```

The Answer:

```python S3 = SymmetricGroup(3) omega = exp(2piI/3)

Standard irreducible representation of S_3

rho_matrices = { S3(()): matrix([[1, 0], [0, 1]]), S3((1,2,3)): matrix([[omega, 0], [0, omega^2]]), S3((1,3,2)): matrix([[omega^2, 0], [0, omega]]), S3((1,2)): matrix([[0, 1], [1, 0]]), S3((1,3)): matrix([[omega^2, 0], [0, omega]]) * matrix([[0, 1], [1, 0]]), S3((2,3)): matrix([[omega, 0], [0, omega^2]]) * matrix([[0, 1], [1, 0]]) }

def rho(g): """Homomorphism from S_3 to GL_2(C)""" return rho_matrices[g]

Verify it's a homomorphism by checking rho(gh) = rho(g)rho(h)

```

Memory Tip:

Remember "CHIRP" - Cube roots, Homomorphism preserves structure, Irreducible means no proper subspaces, Representation uses matrices, Permutations become linear transformations!

The beautiful part is that this representation reveals the "rotational" nature of 3-cycles (using cube roots of unity) and the "reflection" nature of transpositions (swapping coordinates). You're literally seeing the geometric soul of S₃!