Two group actions are equivalent if there is a bijection between the sets and an isomorphism between the groups, that commutes with the action. If this isomorphism is the identity map, then we say that the underlying G-sets are isomorphic. We’ll explore both of these concepts throughout this lecture, using our action graphs visuals and group switchboard analogy. We’ll see why every left action has an equivalent right action, and vice-versa. We’ll see examples of this, including groups acting by multiplication and by conjugation. Finally, we’ll revisit several familiar examples of left vs. right actions involving permutations, now that we have a better framework in which to formalize it. We'll end with the left- and right-permuahedra, which are polytopes that we saw back in Chapter 2.
Course & book webpage (with complete lecture note slides, HW, exams, etc.): https://www.math.clemson.edu/~macaule/visualalgebra/
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0:00 Introduction
1:01 Left vs. right actions
3:37 Action equivalence, informally
5:01 An equivalence between a D₃- and S₃-action
7:12 Two actions on binary squares
9:07 Action equivalence, formally
11:01 Non-isomorphic G-sets from equivalent actions
13:20 Action equivalence (weaker) vs. G-set isomorphism (stronger)
15:19 Left vs. right actions of G on itself by multiplication
19:14 Left vs. right actions of G on itself by conjugation
22:23 Permutation matrices
23:28 Left vs. right actions by permutation matrices
25:11 Left vs. right actions by S₃: swap coordinates vs. swap digits
26:59 Actions by S₄ and the left and right permutahedron