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 W V 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 W V when G is a dihedral group of order 4 N + 2. [ABSTRACT FROM AUTHOR]