Simplicial cell decompositions of ${\mathbb{C}\mathbb{P}}^n$

According to a well-known result in geometric topology, we have \linebreak ${\left({\mathbb{S}}^2\right)}^n\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{/}\mathrm{Sym}(n)={\mathbb{C}\mathbb{P}}^n$, where $\mathrm{Sym}(n)$ 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 $S_3^2$ 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 $\mathrm{Sym}(n)$ 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 $n\ge 4$.