A generalization of Ramanujan's sum over finite groups

Let $G$ be a finite group, and let $x \in G$. Denote by $\langle x^G \rangle$ the normal subgroup of $G$ generated by the conjugacy class of $x$, and define $[x^G] := \{ y \in G : \langle x^G \rangle = \langle y^G \rangle \}$. For any irreducible character $\chi$ of $G$, we define the character sum \begin{align} C_{\chi}(x):= \frac{1}{\chi(\textbf{1})} \sum_{s \in [x^G]} \chi(s),\nonumber \end{align} where $\textbf{1}$ is the identity of $G$. In 1979, Babai proved that $C_{\chi}(x)$ is an eigenvalue of the Cayley graph $\text{Cay}(G, [x^G])$. When $G$ is a finite cyclic group and $x$ is a generator, this sum coincides with the classical Ramanujan sum. Thus, $C_{\chi}(x)$ can be viewed as a generalization of the Ramanujan sum to arbitrary finite groups. A well-known formula for the classical Ramanujan sum involves the M$\ddot{\text{o}}$bius function and Euler's phi function. In this work, we derive an explicit formula for the character sum $C_{\chi}(x)$, which extends the formula of classical Ramanujan sum to the context of finite groups. This generalization not only enrich the theory of Ramanujan's sum but also provide new tools in spectral graph theory, representation theory, and algebraic number theory.