Canonical Root Vectors of SUn Journal of Mathematical Physics, Vol. 8, No. 1, Jan. 1967 Abstract The present paper extends the study of SUn to the determination of particular root vectors and fundamental weights. The root vectors are chosen so as to show explicitly the canonical nesting of subgroups SUn >SUn-1 x U1 >SUn-2 x U1, etc. Explicit matrix representations of the fundamental representations are given. The results are all dependent upon the use of n vectors which define the fundamental region of SUn. Eigenvalues of fundamental invariant operators are also given in terms of these n vectors.