character_theory_posts.txt

# Social Media Posts: Character Theory of Finite Groups
# Generated from: notebooks/published/character_theory.ipynb

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## SHORT-FORM POSTS
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### TWITTER/X (< 280 chars)
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Character theory distills entire group representations into a single function: χ(g) = Tr(ρ(g)). We computed the character table of S₃ and verified orthogonality. Beautiful math in action!

#Python #Math #GroupTheory #AbstractAlgebra

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### BLUESKY (< 300 chars)
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Exploring character theory: how traces of representation matrices encode the structure of finite groups. Built the complete character table of S₃ (symmetric group on 3 elements) and verified the orthogonality relations that make characters so powerful.

#Math #GroupTheory #Python

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### THREADS (< 500 chars)
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Ever wondered how mathematicians compress an entire group representation into a single number?

Character theory does exactly that! The character χ(g) = Tr(ρ(g)) captures essential info about how groups act on vector spaces.

We explored S₃ (all permutations of 3 elements) and found:
- 3 conjugacy classes
- 3 irreducible representations
- A beautiful orthonormality relation: ⟨χᵢ, χⱼ⟩ = δᵢⱼ

The sum of squared degrees equals the group order: 1² + 1² + 2² = 6 = |S₃|

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### MASTODON (< 500 chars)
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Implemented character theory for finite groups in Python!

Key results for S₃:
• Characters are class functions: χ(xgx⁻¹) = χ(g)
• First orthogonality: (1/|G|) Σ χᵢ(g)χⱼ(g)* = δᵢⱼ
• Three irreducible reps: trivial (dim 1), sign (dim 1), standard (dim 2)
• Sum of squares: 1² + 1² + 2² = 6 = |S₃| ✓

The character table is a complete invariant encoding all representation-theoretic information.

#Math #GroupTheory #RepresentationTheory #Python

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## LONG-FORM POSTS
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### REDDIT (r/learnpython or r/math)
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**Title:** Built a Character Table Calculator for Finite Groups in Python - Here's How Characters Reveal Group Structure

**Body:**

I just finished implementing character theory for the symmetric group S₃ and wanted to share what I learned!

**What is character theory?**

A character χ of a group representation ρ is simply the trace of the matrix: χ(g) = Tr(ρ(g)). This single number encodes essential information about how the group acts.

**Why is this cool?**

1. Characters are *class functions* - they're constant on conjugacy classes (elements related by conjugation)
2. Irreducible characters form an orthonormal basis with respect to the inner product ⟨χᵢ, χⱼ⟩ = (1/|G|) Σ χᵢ(g)χⱼ(g)*
3. The number of irreducible representations equals the number of conjugacy classes

**Results for S₃:**

S₃ has 6 elements and 3 conjugacy classes:
- Identity: {e}
- Transpositions: {(12), (13), (23)}
- 3-cycles: {(123), (132)}

The character table:

|  | e | (12) | (123) |
|--|---|------|-------|
| χ_trivial | 1 | 1 | 1 |
| χ_sign | 1 | -1 | 1 |
| χ_standard | 2 | 0 | -1 |

The degrees satisfy: 1² + 1² + 2² = 6 = |S₃|

**Applications:**
- Molecular symmetry in chemistry
- Particle physics (classifying bosons/fermions)
- Generalized Fourier analysis on groups

Check out the full interactive notebook with code and visualizations:
https://cocalc.com/github/Ok-landscape/computational-pipeline/blob/main/notebooks/published/character_theory.ipynb

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### FACEBOOK (< 500 chars)
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Just explored one of the most elegant ideas in abstract algebra: character theory!

The key insight: you can compress an entire group representation (matrices showing how a group acts) into a single function using traces.

For the symmetric group S₃ (all ways to shuffle 3 objects), we found exactly 3 irreducible representations whose squared dimensions sum to 6 - the group's size!

Math is beautiful when patterns like this emerge.

Full notebook: https://cocalc.com/github/Ok-landscape/computational-pipeline/blob/main/notebooks/published/character_theory.ipynb

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### LINKEDIN (< 1000 chars)
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Applying Representation Theory: Building Character Tables with Python

In my latest computational notebook, I explored character theory - a cornerstone of abstract algebra with applications spanning chemistry, physics, and data science.

Key Technical Highlights:

• Implemented group composition and conjugacy class detection algorithms
• Constructed three irreducible representations of S₃: trivial, sign (parity), and the 2D standard representation
• Verified orthogonality relations that make character theory computationally tractable
• Visualized the character table and inner product matrix

The character table of a finite group is a powerful invariant - it encodes:
- Normal subgroup structure (kernels of characters)
- Whether the group is abelian
- Centralizer sizes via orthogonality

Skills demonstrated: NumPy for linear algebra, algorithmic thinking for permutation groups, mathematical rigor in verification.

Character theory underlies techniques in signal processing on graphs, molecular orbital theory, and quantum computing gate synthesis.

View the interactive notebook: https://cocalc.com/github/Ok-landscape/computational-pipeline/blob/main/notebooks/published/character_theory.ipynb

#Mathematics #Python #DataScience #ComputationalThinking #AbstractAlgebra

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### INSTAGRAM (< 500 chars)
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Character Theory of Finite Groups ✨

What you're seeing: The character table of S₃ - a matrix encoding ALL essential information about how this group acts on vector spaces.

The magic:
→ 3 conjugacy classes = 3 irreducible representations
→ 1² + 1² + 2² = 6 (group order!)
→ Characters form an orthonormal basis

The right panel shows perfect orthogonality - different characters are perpendicular, same character has length 1.

Abstract algebra meets Python.

#math #abstractalgebra #grouptheory #python #dataviz #science #coding

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