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arXiv · 2007.05510

McKay Matrices for Finite-dimensional Hopf Algebras

Georgia Benkart·Rekha Biswal·Ellen Kirkman·Van C. Nguyen·Jieru Zhu
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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.

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Georgia Benkart, Rekha Biswal, Ellen Kirkman, Van C. Nguyen, Jieru Zhu. 2020-07-10. McKay Matrices for Finite-dimensional Hopf Algebras. https://arxiv.org/abs/2007.05510

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