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Quantum EngineeringYear 0: Mathematical FoundationsMonth 11Day 288

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Year 0·Month 11·Week 2

Day 288: Group Representations — Groups Acting on Vector Spaces

Day 288 of 2,016~13 min read

Learning Objectives

  • •Define group representations on vector spaces
  • •Construct matrix representations in a given basis
  • •Understand equivalent representations
  • •Identify the trivial, regular, and permutation representations
  • •Connect representations to quantum state transformations
  • •Implement representations computationally

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OverviewPrerequisitesLearning Objectives1 Definition of RepresentationThe Fundamental DefinitionTerminologyMatrix Representations2 Basic ExamplesExample 1 The Trivial RepresentationExample 2 The Sign Representation of S_nExample 3 The Defining RepresentationExample 4 The Regular RepresentationExample 5 Cyclic Group mathbbZ_n3 Equivalent RepresentationsDefinitionIn Matrix TermsWhy Equivalence Matters4 Operations on RepresentationsDirect SumTensor ProductDual RepresentationComplex Conjugate5 Quantum Mechanics ConnectionStates as Representation SpacesWhy Representations Matter in QMThe Fundamental PostulateExample Rotations6 Worked ExamplesExample 1 Representations of mathbbZ_2Example 2 Representations of S_3Example 3 Matrix Representation of D_37 Computational Lab8 Practice ProblemsProblem Set A Basic RepresentationsProblem Set B Equivalence and OperationsProblem Set C Quantum Applications9 SummaryKey DefinitionsKey Examples10 Preview Day 289
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