Permutation polynomials, projective polynomials, and bijections between $\mu_{\frac{q^n-1}{q-1}}$ and $PG(n-1,q)$

Using arbitrary bases for the finite field $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}$, we obtain the generalized M\"obius transformations (GMTs), which are a class of bijections between the projective geometry $PG(n-1,q)$ and the set of roots of unity $\mu_{\frac{q^n-1}{q-1}}\subseteq\mathbb{F}_{q^n}$, where $n\geq 2$ is any integer. We also introduce a class of projective polynomials, using the properties of which we determine the inverses of the GMTs. Moreover, we study the roots of those projective polynomials, which lead to a three-way correspondence between partitions of $\mathbb{F}_{q^n}^\ast,\mu_{\frac{q^n-1}{q-1}}$ and $PG(n-1,q)$. Through this correspondence and the GMTs, we construct permutation polynomials of index $\frac{q^n-1}{q-1}$ over $\mathbb{F}_{q^n}$.