Given two strictly upper triangular matrices X, Y ∈ Cm×m, we study the range WY (X) = {trnXn-1Y* : n ∈ N}, where N is the group of unit upper triangular matrices in Cm×m. We prove that it is either a point or the whole complex plane. We characterize when it is a point. We also obtain some convexity result for a similar range, where N is replaced by any ball of Ck(k = m(m - 1)/2) embedded in N , m = 4.