Nonzero coefficients in restrictions and tensor products of supercharacters of Un(q)

The standard supercharacter theory of the finite unipotent upper-triangular matrices Un(q) gives rise to beautiful combinatorics based on set partitions. As with the representation theory of the symmetric group, embeddings of Um(q)⊆Un(q) for m⩽n lead to branching rules. Diaconis and Isaacs established that the restriction of a supercharacter of Un(q) is a nonnegative integer linear combination of supercharacters of Um(q) (in fact, it is polynomial in q). In a first step towards understanding the combinatorics of coefficients in the branching rules of the supercharacters of Un(q), this paper characterizes when a given coefficient is nonzero in the restriction of a supercharacter and the tensor product of two supercharacters. These conditions are given uniformly in terms of complete matchings in bipartite graphs.