If \(M\) is a compact, connected, oriented \(3\)-manifold with boundary, and \(\widehat M\) is the space obtained from \(M\) by collapsing each boundary component to a point, then a triangulation \(\mathcal F\) of \(M\) is an identification of \(\widehat M\) with a closed \(3\)-cycle, i.e. a space obtained from a collection of simplices by gluing together pairs of faces via affine homeomorphisms. If \(z_j\) are complex variables, one for each simplex \(\Delta_j\) of a triangulation \(\mathcal F\), and assigned to the edges of \(\Delta_j\) are shape parameters \(z_j\), \(z_j'=\frac{1}{1-z_j}\), and \(z''_j=1-\frac{1}{z_j}\), then the Thurston’s gluing equations for each \(1\)-cell of \(\mathcal F\) are \(\prod\limits_j(z_j)^{A_{e,j}}\prod\limits_j(1-z_j)^{B_{e,j}}=\varepsilon_e\), where \(A\) and \(B\) are integer matrices whose columns are parametrized by the simplices of \(\mathcal F\) and \(\varepsilon_e\in\{-1,1\}\). Each non-degenerate solution \(z_j\) explicitly determines a representation of \(\pi_1(M)\) in \(\mathrm{PGL}(2,\mathbb C)=\mathrm{PSL}(2,\mathbb C)\). The matrices \(A\) and \(B\) have some remarkable symplectic properties that play a fundamental role in exact and perturbative Chern-Simons theory for \(\mathrm{PSL}(2,\mathbb C)\). In [Algebr. Geom. Topol. 15, No. 1, 565–622 (2015; Zbl 1347.57014)], S. Garoufalidis et al. generalized Thurston’s gluing equations to representations in \(\mathrm{PGL}(n,\mathbb C)\), i.e. they constructed a system of equations such that each solution determines a representation of \(\pi_1(M)\) in \(\mathrm{PGL}(n,\mathbb C)\). It seems that the \(\mathrm{PGL}(n,\mathbb C)\)-gluing equations play a similar role in \(\mathrm{PGL}(n,\mathbb C)\)-Chern-Simons theory as Thurston’s gluing equations play in \(\mathrm{PSL}(2,\mathbb C)\)-Chern-Simons theory. In [Topology ’90, Contrib. Res. Semester Low Dimensional Topol., Columbus/OH (USA) 1990, Ohio State Univ. Math. Res. Inst. Publ. 1, 243–271 (1992; Zbl 0768.57006)], W. D. Neumann has proved some symplectic properties of Thurston’s gluing equations that play an important role in recent developments of exact and perturbative Chern-Simons theory. Some symplectic properties of the gluing equations were encoded in terms of a chain complex \(\mathcal J=\mathcal{J(F)}\). In this paper, the authors prove similar symplectic properties of the \(\mathrm{PGL}(n,\mathbb C)\)-gluing equations for all ideal triangulations of compact oriented 3-manifolds.