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Quantum Engineering Year 0: Mathematical Foundations Month 11Day 286

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Year 0·Month 11·Week 1

Day 286: Permutation Groups — The Symmetric Group

Day 286 of 2,016~16 min read

Learning Objectives

•Write permutations in cycle notation
•Compute products of permutations and cycles
•Determine the order of a permutation from its cycle structure
•Classify permutations as even or odd
•Work with the alternating group $A_n$
•Connect permutation groups to quantum particle exchange

Today's Schedule (7 hours)

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Overview Prerequisites Learning Objectives 1 The Symmetric Group Definition Two-Line Notation Composition 2 Cycle Notation Cycles Disjoint Cycle Decomposition Cycle Type 3 Properties of Permutations Order of a Permutation Transposition Decomposition Sign of a Permutation 4 The Alternating Group Definition Properties Generators of A_n Simplicity of A_n 5 Cayleys Theorem 6 Quantum Mechanics Connection Identical Particles Bosons and Fermions Pauli Exclusion Principle Slater Determinants Symmetrization and Antisymmetrization 7 Worked Examples Example 1 Cycle Decomposition Example 2 Computing Products Example 3 Conjugacy Classes in S_4 8 Computational Lab 9 Practice Problems Problem Set A Basic Computations Problem Set B Structure Problem Set C Advanced 10 Summary Key Formulas Key Theorems 11 Preview Day 287

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