Admittedly the shirt genome has yet to be mapped, and the heritability of sleeve length is not widely accepted in either the textile or molecular biology community.
That doesn't mean there are no genes being fiddled with in the making of that oxford cloth button down. Genetically modified cotton, also known as Bt, or transgenic, cotton, is grown all over the world and is present in unknown numbers and styles of garments.
Many consumers want food that is made from GM plants labelled at the very least, so they know what they are getting. And yet, what is more personal? Corn? Or shirts? Well, perhaps corn.