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Quantum Engineering Year 0: Mathematical Foundations Month 11Day 293

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Year 0·Month 11·Week 2

Day 293: Representations of the Symmetric Group

Day 293 of 2,016~13 min read

Learning Objectives

•Classify irreps of $S_n$ using partitions and Young diagrams
•Construct Young tableaux and understand their role
•Apply the hook length formula for dimensions
•Connect $S_n$ representations to identical particle physics
•Understand symmetrizers and antisymmetrizers
•Use Young tableaux in quantum mechanics applications

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Overview Prerequisites Learning Objectives 1 Partitions and Young Diagrams Partitions Young Diagrams Conjugate Partition 2 Correspondence with Irreps The Fundamental Theorem Special Cases Dimension from Hook Lengths 3 Young Tableaux Standard Young Tableaux Semistandard Young Tableaux 4 Constructing Irreps Young Symmetrizers The Standard Module 5 Quantum Mechanics Connection Identical Particles Parastatistics Theoretical The Spin-Statistics Theorem Multi-Electron Wave Functions Young Tableaux and Coupling 6 Worked Examples Example 1 Irreps of S_3 Example 2 Irreps of S_4 Example 3 Two-Electron System 7 Computational Lab 8 Practice Problems Problem Set A Partitions and Dimensions Problem Set B Young Tableaux Problem Set C Quantum Mechanics 9 Summary Key Results Dimension Table Small n 10 Preview Day 294

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