McKay Matrices for Pointed Rank One Hopf Algebras of Nilpotent Type

Let H be a finite-dimensional pointed rank one Hopf algebra of nilpotent type over a finite group G. In this paper, we investigate the McKay matrix W-V of H for tensoring with the 2-dimensional indecomposable H-module V:= M ( 2,0 ) . It turns out that the characteristic polynomial, eigenvalues and eigenvectors of WV are related to the character table of the finite group G and a kind of generalized Fibonacci polynomial. Moreover, we construct some eigenvectors of each eigenvalue for W-V by using the factorization of the generalized Fibonacci polynomial. As an example, we explicitly compute the characteristic polynomial and eigenvalues of W-V and give all eigenvectors of each eigenvalue for WV when G is a dihedral group of order 4N +2 .