Plateaued functions on finite abelian groups and partial geometric difference sets

As a generalization of plateaued functions on finite fields and bent functions (perfect nonlinear functions) on finite abelian groups, plateaued functions on finite abelian groups were introduced in [B. Xu, Plateaued functions, partial geometric difference sets, and partial geometric designs, J. Combin. Des. 27 (2019), 756-783]. In this paper, we continue the research in the paper mentioned above. We will obtain various characterizations of plateaued functions; these characterizations establish close connections between plateaued functions and some combinatorial objects: partial geometric difference sets and related partial geometric difference families. Then we introduce the complementary matrix and Cayley matrix for a subset of a finite group and use them to characterize partial geometric difference sets. As applications, we will show how to construct directed strongly regular graphs from partial geometric difference sets and establish a natural relation between partial geometric difference sets and partial geometric designs. The tensor product of a group algebra and a cyclotomic field is an important tool for our discussions.