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Journal Article

McKay matrices for finite-dimensional Hopf algebras


Biswal,  Rekha
Max Planck Institute for Mathematics, Max Planck Society;

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Benkart, G., Biswal, R., Kirkman, E., Nguyen, V. C., & Zhu, J. (2022). McKay matrices for finite-dimensional Hopf algebras. Canadian Journal of Mathematics, 74(3), 686-731. doi:10.4153/S0008414X21000067.

Cite as: https://hdl.handle.net/21.11116/0000-000A-1582-B
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.