Explicit construction of exact unitary designs

The purpose of this paper is to give explicit constructions unitary $t$-designs in the group $U(d)$ for all $t$ and $d$. It seems that were so far known only very special cases. Here construction means entries matrices are given by values elementary functions at root some polynomials. We will discuss what best such $4$-designs $U(4)$ obtained these methods. Indeed we an inductive designs on compact groups using Gelfand pairs $(G,K)$. Note $(U(n),U(m) \times U(n-m))$ a pair. By zonal spherical $(G,K)$, can construct $G$ from $K$. remark our proofs use representation theory crucially. also method be applied orthogonal $O(d)$, thus provides another $d$ dimensional sphere $S^{d-1}$ induction