McKay Matrices for Pointed Rank One Hopf Algebras of Nilpotent Type

Abstract

Let [Formula: see text] be a finite-dimensional pointed rank one Hopf algebra of nilpotent type over a finite group [Formula: see text]. In this paper, we investigate the McKay matrix [Formula: see text] of [Formula: see text] for tensoring with the 2-dimensional indecomposable [Formula: see text]-module [Formula: see text]. It turns out that the characteristic polynomial, eigenvalues and eigenvectors of [Formula: see text] are related to the character table of the finite group [Formula: see text] and a kind of generalized Fibonacci polynomial. Moreover, we construct some eigenvectors of each eigenvalue for [Formula: see text] by using the factorization of the generalized Fibonacci polynomial. As an example, we explicitly compute the characteristic polynomial and eigenvalues of [Formula: see text] and give all eigenvectors of each eigenvalue for [Formula: see text] when [Formula: see text] is a dihedral group of order [Formula: see text].