Relation between spectra of a Cayley graph of a group and irreducible characters of that group I know the following fact:
If $G$ is an abelian group and $S\subset G$ be a subset of G such that $1\notin G$ and $S=S^{-1}$ and we draw an edge between $g$ and $h$ if and only if $hg^{-1}\in S$,then eigenvalues  of the adjacency matrix of this graph are exactly given by $\sum_{s\in S}\ \chi_i(s)$ where $\chi_i's$ are irreducible characters of the group $G$.
Now, I was wondering about how far this is true for non-abelian groups. I did some computation with $S_3$ by taking $S$ to be $\{(12),(13),(23)\}$. There I got the eigenvalues $\sum_{s\in S}\ \chi_i(s)$  with multiplicity exactly $d_i^2$, where $d_i=\dim(\chi_i)$. Is this true in general?
Can anyone please help me in this direction or give any reference to some literature where these kind of things are done?
Thanks in advance.
 A: I'll elaborate on Eberhard's comment. Let's work in the realm of finite groups $G$, otherwise we'll have to contend with serious issues coming from operator theory. Inside of our finite group $G$ take a subset $S$ which: (1) generates $G$, (2) is a union of conjugacy classes, and (3) is closed under inversion. The prototypical example is when $G$ is a finite reflection group (like $S_n$) and $S$ is the set of all reflections.
We can define an (unoriented) Cayley graph structure on $G$ with respect to $S$, i.e. a graph whose vertices are indexed by group elements $g \in G$ and where an edge is drawn between $g_1, g_2 \in G$ whenever there exists $s \in S$ such that $g_2 = s g_1$. By property (3) the edge-drawing convention is reflexive (otherwise we'd have to settle for an oriented graph) and property (1) insures that the graph is connected (although you may choose to dispense with this feature if not needed). Regarding property (2): Instead of the adjacency matrix/operator $A_S$, let's consider the graph Laplacian $\Delta_S : \Bbb{C}[G] \rightarrow \Bbb{C}[G]$ defined by
\begin{equation}
\Delta_S \varphi \, (g) \, := \ \varphi(g) \, - \, {1 \over {|S|}}\sum_{s \in S} \, \varphi(sg)
\end{equation}
where $\Bbb{C}[G]$ is the vector space of $\Bbb{C}$-valued functions $\varphi: G \rightarrow \Bbb{C}$ on the group. Note that $A_S = |S| \cdot (\Bbb{1} - \Delta_S)$ so solving one eigenfunction/eigenvalue problem solves the other. Property (2) insures that $\Delta_S \varphi$ is a class-function whenever $\varphi$ is.
Now take an irreducible, complex representation $\varrho$ of $G$ and consider its character $\chi_\varrho$ (which we can view as a class-function). Then $\Delta_S \chi_\varrho$ must also be a class-function. Indeed
it's an eigenfunction of $\Delta_S$ with eigenvalue
\begin{equation}
{\Delta_S \chi_\varrho (1) \over {\dim(\varrho)}} 
\ = \ 1 \, - \, {1 \over {|S| \, \dim(\varrho)}} \, \sum_{s \in S}
\chi_\varrho(s)
\end{equation}
the righthand side of which can be calculated, for example, using the Murnaghan–Nakayama rule in the case of the symmetric group $G = S_n$. This agrees with your calculation for the  adjacency matrix (also noting Eberhard's correction).
This eigenvector/eigenvalue calculation follows from the observation that $\Delta_S$ is in fact a convolution operator, i.e.
\begin{equation}
\Delta_S \varphi \ = \ \Big( \delta_1 \, - \, {1 \over {|S|}} \delta_S \Big) * \varphi
\end{equation}
where $\delta_1$ and $\delta_S$ are the delta functions supported on $\{1\}$ and $S$ respectively,
together with character-orthogonality, i.e.
\begin{equation}
\chi_\varrho * \chi_\sigma
\ = \ \left\{
\begin{array}{cl}
{|G| \over {\dim(\varrho)}} \chi_\varrho
&\text{if $\varrho \cong \sigma$} \\
0 &\text{otherwise}
\end{array}
\right.
\end{equation}
for any pair of irreducible, complex representations
$\varrho, \sigma$ of $G$.
I emphasize the graph Laplacian (instead of the adjacency matrix) in order to stike a parallel with the theory of compact Lie groups $G$, where it is known (essentially by the Peter-Weyl theorem) that the irreducible characters of $G$ are exactly the  eigenfunctions of the $G$-invariant Beltrami-Laplace operator $\Delta$ (i.e. the Casimir operator) when restricted to the space of class functions.
ines.
