The spectral decomposition of graph adjacency matrices is an essential ingredient in the design of graph signal processing (GSP) techniques. When the adjacency matrix has multi-dimensional eigenspaces, it is desirable to base GSP constructions on a particular eigenbasis that better reflects the graph's symmetries. In this paper, we provide an explicit and detailed representation-theoretic account for the spectral decomposition of the adjacency matrix of a weighted Cayley graph. Our method applies to all weighted Cayley graphs, regardless of whether they are quasi-Abelian, and offers detailed descriptions of eigenvalues and eigenvectors derived from the coefficient functions of the representations of the underlying group. Next, we turn our attention to constructing frames on Cayley graphs. Frames are overcomplete spanning sets that ensure stable and potentially redundant systems for signal reconstruction. We use our proposed eigenbases to build frames that are suitable for developing signal processing on Cayley graphs. These are the Frobenius--Schur frames and Cayley frames, for which we provide a characterization and a practical recipe for their construction.

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