On the dimension of matrix embeddings of torsion-free nilpotent groups

On the other hand, we also prove a general exponential upper bound on the dimension of the embedding by showing that for any torsion free, finitely generated nilpotent group the matrix representation produced by Nickel's algorithm has never larger dimension than Jennings' embedding. Moreover, when starting with a special Mal'cev basis, Nickel's embedding for UTn(Z) has only quadratic size. Finally, we consider some special cases like free nilpotent groups and Heisenberg groups and compare the sizes of the embeddings.