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Mathematics, Rings and Algebras, Representation Theory
Abstract:
For a finite-dimensional Hopf algebra $A$, the McKay matrix $M_V$ of an
$A$-module $V$ encodes the relations for tensoring the simple $A$-modules with
$V$. We prove results about the eigenvalues and the right and left
(generalized) eigenvectors of $M_V$ by relating them to characters. We show how
the projective McKay matrix $Q_V$ obtained by tensoring the projective
indecomposable modules of $A$ with $V$ is related to the McKay matrix of the
dual module of $V$. We illustrate these results for the Drinfeld double $D_n$
of the Taft algebra by deriving expressions for the eigenvalues and
eigenvectors of $M_V$ and $Q_V$ in terms of several kinds of Chebyshev
polynomials. For the matrix $N_V$ that encodes the fusion rules for tensoring
$V$ with a basis of projective indecomposable $D_n$-modules for the image of
the Cartan map, we show that the eigenvalues and eigenvectors also have such
Chebyshev expressions.
