By Schur's lemma the unitary π(g) is unique up to multiplication by a scalar ζ with |ζ| = 1, so that π defines a projective unitary representation of Sp(n). Representatives can be chosen for π(g), unique up to a sign, which show that the 2-cocycle for the projective representation of Sp(2 n, R) takes values ±1. In fact elements of the group Sp(n, R) are given by 2 n × 2 n real matrices g satisfying