According to a well-known result in geometric topology, we have \(\left( \mathbb {S}^2 \right) ^{n}\!\!/\operatorname {Sym}(n) = \mathbb{C}\mathbb{P}^{n}\) , where \(\operatorname {Sym}(n)\) acts on \(\left( \mathbb {S}^2 \right) ^{n}\) by coordinate permutation. We use this fact to explicitly construct a regular simplicial cell decomposition of \(\mathbb{C}\mathbb{P}^{n}\) for each \(n \ge 2\) . In more detail, we start with the standard two triangle crystallisation \(S^2_3\) of the 2-sphere \(\mathbb {S}^2\) , in its n-fold Cartesian product. We then construct a simplicial subdivision of this product and prove that the \(\operatorname {Sym}(n)\) quotient of this subdivision yields a simplicial cell decomposition of \(\mathbb{C}\mathbb{P}^n\) . The first derived subdivision of this cell complex is a simplicial triangulation of \(\mathbb{C}\mathbb{P}^n\) . To the best of our knowledge, this is the first explicit description of triangulations of \(\mathbb{C}\mathbb{P}^n\) for \(n \ge 4.\)
