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Quantum Engineering Year 0: Mathematical Foundations Month 11Day 289

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Year 0·Month 11·Week 2

Day 289: Reducible and Irreducible Representations

Day 289 of 2,016~14 min read

Learning Objectives

•Define invariant subspaces and identify them
•Distinguish reducible from irreducible representations
•State and apply Maschke's theorem
•Decompose representations into irreducible components
•Connect irreducibility to quantum mechanical degeneracy
•Implement computational tests for reducibility

Today's Schedule (7 hours)

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Overview Prerequisites Learning Objectives 1 Invariant Subspaces Definition Examples of Invariant Subspaces Finding Invariant Subspaces 2 Reducible and Irreducible Representations Definitions Block Diagonal Form 3 Maschkes Theorem Statement Proof Idea Significance 4 Counting Irreducible Representations For Finite Groups Examples 5 Finding Irreducible Components The General Strategy Using Characters Preview Projection Operators 6 Quantum Mechanics Connection Irreps and Degeneracy Multiplets Selection Rules Symmetry Breaking 7 Worked Examples Example 1 Decompose the 3D Permutation Rep of S_3 Example 2 Decompose Regular Rep of mathbbZ_3 Example 3 Test Irreducibility of 2-dim Rep 8 Computational Lab 9 Practice Problems Problem Set A Invariant Subspaces Problem Set B Decomposition Problem Set C Quantum Applications 10 Summary Key Concepts Key Theorems 11 Preview Day 290

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