Contents Lapter 1. Introduction 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10. Preliminaries The case of commuting pairs in a simply connected group c-pairs Commuting triples C-triples Quotients of diagram automorphisms Description of S(k) and S (g, k) Chern-Simons invariants and Witten's "Clockwise Symmetry Conjecture" Outline of the paper History 1 2 4 4 5 6 7 9 11 13 Chapter 2. Almost commuting Af-tuples 15 2.1. An invariant for almost commuting N-tuples 15 2.2. The case of rank zero 16 2.3. The case of arbitrary rank 18 Chapter 3. Some characterizations of groups of type A 21 3.1. Generalities on subroot systems 21 3.2. Action of CG on an alcove 21 3.3. A first characterization of groups of type A 24 3.4. Subgroups associated with elements of the center 25 3.5. A further characterization of products of groups of type A 26 3.6. A consequence of Proposition 3.5.1 27 3.7. Application to generalized Cartan matrices and affine diagrams 28 3.8. Numerology of clockwise symmetry 30 Chapter 4. c-pairs 35 4.1. The rank zero case 35 4.2. The general case 37 Chapter 5. Commuting triples 39 5.1. Commuting triples of rank zero 39 5.2. A list of all simple groups with rank zero commuting triples 41 5.3. Action of the outer automorphism group of G 41 5.4. Action of the center of G 41 5.5. The general case 42

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