Coalitional games, including Coalition Structure Generation (CSG), have been attracting considerable attention from the AI research community. Traditionally, the input of a coalitional game is a black-box function called a characteristic function. Previous studies have found that many problems in coalitional games tend to be computationally intractable in this black-box function representation. Recently, several concise representation schemes for a characteristic function have been proposed. Among them, a synergy coalition group (SCG) has several good characteristics, but its representation size tends to be large compared to other representation schemes. We propose a new concise representation scheme for a characteristic function based on a Zero-suppressed Binary Decision Diagram (ZDD) and a SCG. We show our scheme (i) is fully expressive, (ii) can be exponentially more concise than the SCG representation, (iii) can solve core-related problems in polynomial time in the number of nodes, and (iv) can solve a CSG problem reasonably well by utilizing a MIP formulation. A Binary Decision Diagram (BDD) has been used as unified infrastructure for representing/manipulating discrete structures in such various domains in AI as data mining and knowledge discovery. Adapting this common infrastructure brings up the opportunity of utilizing abundant BDD resources and cross-fertilization with these fields.