Action of wreath product on function space
Suppose both and are being viewed along with specific group actions of each, i.e., they are both being viewed as groups of permutations, with acting on a set and acting on a set . Then, comes equipped with a natural action on the set of all functions from to .
First, recall that is the external semidirect product . To specify an action of this, we will specify how and act; then, via the equivalence of internal and external semidirect product, we would have defined the action of a generating set and hence of all of . Here is the action:
- acts on as follows: Given a function and a function , define as the function given by , where we use the action of on . One way of thinking of this is that the action is coordinate-wise or point-wise.
- acts on as follows: Given an element and a function , we define as the map where we use the action of on on the inside.