Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

The entire bottom block in figure 1 can be considered to illustrate this direct product. Notice that the control group action of $G(C)$ on $C$ induces an action of $G(C)$ on the set of indexes $c$ within the direct product in (4). Most crucially, this action of $G(C)$ is an automorphic action on the direct product.

Next, take the semi-direct product of the entire lower block and the control group above, thus:

prod { G(F )c} sO G(C). c (- C

(5)

The lower block $prod c (- C G(F)c$ is the normal subgroup of the semi-direct product. In any semi-direct product, the upper group acts as an automorphism group of the normal subgroup (here the lower block); and in this case the chosen automorphic action will be the one defined in the previous paragraph.

Next consider the set $F × C$ , which we will call the data set. Notice that this set decomposes into the fiber-set copies $Fc$ .

Now for the final fundamental point concerning wreath products: There is a group action of the wreath product $G(F )OwG(C)$ on the data set $F × C$ . To define this action, let us assume, only for the purposes of notation, that the control set $C$ is finite, of cardinality $n$ . Observe that, by the semi-direct product structure of the wreath product (as shown in expression (5)), a single element from the wreath product must be of the form:

< ( Tc1 , Tc2 , ... , Tcn )|g >

(6)

where each $T ci$ is an element taken from its fiber-group copy $G(F ) ci$ (column in figure 1); and $g$ is an element taken from the control group (upper level in figure 1).

Let us therefore see the effect of the full element in expression (6) on a single point $(f , c ) i$ in the data set $F × C$ . Notice, since each $T cj$ in expression (6) acts only on its personal fiber-set copy, only the element $T ci$ will act on the point $(f , c ) i$ . It sends it to the point $(T f , c ) ci i$ . Finally, the element $g$ in expression (6) moves this point to its corresponding position in the fiber indexed by $gc i$ .

The action will be called the full wreath action, and we have seen that it is given thus:

{ G(F)wOG(C) × [F × C] - --> [F × C] ( < ( Tc1 , Tc2 , ... , Tcn )| g > , (f,ci) ) '- --> (Tcif,gci).

(7)

7 Mathematical Theory of Transfer

We are now ready to give our rigorous theory of transfer, as follows: Each copy $G(F )c$ , of the fiber group, acts on its own copy of the fiber set $Fc$ . One can view the control group as transferring the fiber-group copies around the fiber-set copies. In fact, this action is achieved by the automorphic action of the control group within the wreath product, as given by the map $t$ . This action sends the fiber-group copies onto each other via conjugation. Therefore: