We adress the representation theory of the infinite unitriangular group U∞ as an analogue to the n × n unitriangular group Un. A family of representatations is defined and the main properties are examined. Since U∞ is the direct limit of the family {Un}, such representations yelds a Supercharacter theory that is aproximated by the supercharacters of all Un. This group serves as a prototype to other infinite discrete groups.
