We introduce coordinates on the spaces of framed and decorated representations of the fundamental group of a surface with boundary into the symplectic group Sp(2n,R). These coordinates provide a noncommutative generalization of the parameterizations of the spaces of representations into SL(2,R) or PSL(2,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.
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