Gauss sums of some matrix groups over $\Bbb Z/n\Bbb Z$

Abstract: In this paper, we will explicitly calculate Gauss sums for the general linear groups and the special linear groups over $\Bbb Z_n$, where $\Bbb Z_n=\Bbb Z/n \Bbb Z$ and $n>0$ is an integer. For $r$ being a positive integer, the formulae of Gauss sums for ${\rm GL}_r(\Bbb Z_n)$ can be expressed in terms of classical Gauss sums over $\Bbb Z_n$, while the formulae of Gauss sums for ${\rm SL}_r(\Bbb Z_n)$ can be expressed in terms of hyper-Kloosterman sums over $\Bbb Z_n$. As an application, we count the number of $r\times r$ invertible matrices over $\Bbb Z_n$ with given trace by using the the formulae of Gauss sums for ${\rm GL}_r(\Bbb Z_n)$ and the orthogonality of Ramanujan sums.