讲座内容：

By the induced character theory, we first establish the irreducible decomposition of a permutation representation. Then, using it as a unified tool, we derive a decomposition formula for the distance spectrum and eigenspace of a regular lift of a graph, the adjacency spectra and eigenspaces of a relative lift and a ramified uniform lift of a digraph. The latter results are    complements of [C. Dalf$\acute{\text{o}}$, M.A. Fiol, S. Pavl\'{i}kov\'a and J. $\check{\text{S}} $ir$\acute{\text{a}}\check{\text{n}}$, Spectra and eigenspaces of arbitrary lifts of graphs, J. Algebraic Combin. 54 (2021) 651-672] and [A.P. Deng, I. Sato, Y.K. Wu, Characteristic polynomials of ramified uniform covering digraphs, European J. Combin. 28 (2007) 1099-1114].