  We introduce coordinates on the spaces of framed and decorated
representations of the fundamental group of a surface with nonempty boundary
into the symplectic group $Sp(2n,\mathbf R)$. These coordinates provide a
noncommutative generalization of the parametrizations of the spaces of
representations into $SL(2,\mathbf R)$ or $PSL(2,\mathbf R)$ given by Thurston,
Penner, Kashaev, and Fock-Goncharov. On the space of decorated symplectic
representations the coordinates give a geometric realization of the
noncommutative cluster-like structures introduced by Berenstein-Retakh. The
locus of positive coordinates maps to the space of framed maximal
representations. We use this to determine an explicit homeomorphism between the
space of framed maximal representations and a quotient by the group $O(n)$.
This allows us to describe the homotopy type and, when $n=2$, to give an exact
description of the singularities. Along the way, we establish a complete
classification of pairs of nondegenerate quadratic forms.
