As mentioned previously, for the situation regarding minimal periodic colorings of is more complicated. The assignment
There are also essentially three inequivalent periodic 4-colorings with obtained in this way, corresponding to
For larger values of it is impractical to catalog the coherent color shift groups in silico via brute force: either a reasonably sophisticated algorithm or some (presumably nontrivial representation-theoretic) calculation in papyro is wanted. Nevertheless, it is easy to see that a cyclic coloring of exists for all (and hence that for prime the cyclic coloring is the only periodic -coloring): simply let
The cyclic coloring is obtained by setting
Color quotients and the cyclic color space
For we may consider a -coloring of and the attendant color quotient , where . The color projection map is defined in the obvious way, and we may identify and with only a slight abuse of notation. In particular, if is the cyclic coloring, the resulting color quotient is called the cyclic color space and denoted
Consider a càdlàg random walk on , for , and write for the projected walk on Let denote the th jump time of (and hence also ), and in a further abuse of notation write Let be defined via Now the color shift from to is
Without loss of generality, let and . Define
Note that , where indicates the cyclic color shift action. Now
Therefore , and we have that the color shift from to is
We proceed along similar lines to compute from and
This equation illustrates the ease and utility of projecting random walks from to It is also straightforward to show that for the assignment
produces the cyclic coloring.
In a followup post I’ll sketch how to identify the coherent color shift groups of low order and hint at some of the ways in which this construction might be applied.