  The standard supercharacter theory of the finite unipotent upper-triangular
matrices $U_n(q)$ gives rise to a beautiful combinatorics based on set
partitions. As with the representation theory of the symmetric group,
embeddings of $U_m(q)\subseteq U_n(q)$ for $m\leq n$ lead to branching rules.
Diaconis and Isaacs established that the restriction of a supercharacter of
$U_n(q)$ is a nonnegative integer linear combination of supercharacters of
$U_m(q)$ (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 $U_n(q)$, 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.
