Visual Algebra, Lecture 5.8: Simply transitive actions скачать в хорошем качестве

Visual Algebra, Lecture 5.8: Simply transitive actions

A group action, or G-set it defines, is transitive if its action graph is connected. An action is free if the stabilizer of every element is trivial. This means that it is as “uncollapsed as possible.” Actions or G-sets that are transitive and free are called simply transitive. In this lecture, we’ll see that every simply transitive action is isomorphic to a group acting on itself by multiplication. In other words, the simply transitive action graphs are precisely the Cayley graphs. Then, we’ll see a number of examples of simply transitive actions, mostly from tilings of polygons, polytopes, the xy-plane, and even the hyperbolic plane. Course & book webpage (with complete lecture note slides, HW, exams, etc.): https://www.math.clemson.edu/~macaule... ------------------------------------------------------------------------------------------------------------------------------------------------------ 0:00 Introduction 1:03 Transitive, free, and faithful actions 3:32 An action that is free but not transitive 4:55 Examples of simply transitive actions 6:24 Every simply transitive action graph is a Cayley graph 12:23 A simply transitive action of ℤ×ℤ 14:00 Simply transitive actions of D₃ and D₄ 18:18 From finite to affine reflection groups 19:03 The affine Weyl group of type Ã₂ 21:44 The affine Weyl group of type C̃₂ 25:10 Weyl groups and Dynkin diagrams 28:31 The affine Weyl group of type G̃₂ 29:54 Coxeter groups and affine Weyl groups of higher rank 33:52 Hyperbolic Coxeter groups 36:15 A simply transitive action of PSL₂(ℤ)