Hokum and Guinea are exactly right.
At bare minimum, the chickens will bully the pigeon endlessly.
The worst case scenario is that they will eventually kill the pigeon and cannibalize it.
I have seen many situations where pigeons were kept with chickens and even where they were given open freedom to come and go as they like, it was not unusual to see the chickens get into the pigeon nests to attack and kill the squabs. Often, you'd find the squabs dead on the floor and torn to pieces.
Another thing I would never do is keep pigeons and doves together, unless you had A LOT of space for only a few birds. In closed quarters, the end result is usually scalped doves. You MIGHT get away with it if they were all weaned together as youngsters. But adults, almost never.
I once kept some Bobwhites with some pigeons in a pen together out of necessity and that seemed to work well, except if the quail got spooked, they created a lot of pandemonium.
I have 60 Jumbo Cotournix eggs in the incubator right now and would not even dream of housing them with my pigeons.
As a rule, I do not even house other breeds of pigeons together unless it's by intention.
Get a mate for your Fantail and give them their own place. They will give you plenty of pleasure.
When I was a kid, I had a whole bunch of Tailmarked and Body Marked Fantails and used to let them out for a few hours each day to get baths and to pick up a little green food. They were well marked, but in no way show quality. I still remember those Fantails vividly and there was just something very relaxing and pleasant about watching them outside. It still sticks with me all these years later.
